Tiling Solver

Tiling a n-dimensional polyomino with n-dimensional polyominoes.

This module defines two classes:

• sage.combinat.tiling.Polyomino class, to represent polyominoes in arbitrary dimension. The goal of this class is to return all the rotated, reflected and/or translated copies of a polyomino that are contained in a certain box.

• sage.combinat.tiling.TilingSolver class, to solve the problem of tiling a \(n\)-dimensional polyomino with a set of \(n\)-dimensional polyominoes. One can specify if rotations and reflections are allowed or not and if pieces can be reused or not. This class convert the tiling data into rows of a matrix that are passed to the DLX solver. It also allows to compute the number of solutions.

This uses dancing links code which is in Sage. Dancing links were originally introduced by Donald Knuth in 2000 [Knuth1]. Knuth used dancing links to solve tilings of a region by 2d pentaminoes. Here we extend the method to any dimension.

In particular, the sage.games.quantumino module is based on the Tiling Solver and allows to solve the 3d Quantumino puzzle.

AUTHOR:

• Sébastien Labbé, June 2011, initial version

• Sébastien Labbé, July 2015, count solutions up to rotations

• Sébastien Labbé, April 2017, tiling a polyomino, not only a rectangular box

EXAMPLES:

2d Easy Example

Here is a 2d example. Let us try to fill the \(3 \times 2\) rectangle with a \(1 \times 2\) rectangle and a \(2 \times 2\) square. Obviously, there are two solutions:

Sage

sage: from sage.combinat.tiling import TilingSolver, Polyomino
sage: p = Polyomino([(0,0), (0,1)])
sage: q = Polyomino([(0,0), (0,1), (1,0), (1,1)])
sage: T = TilingSolver([p,q], box=[3,2])
sage: it = T.solve()
sage: next(it)
[Polyomino: [(0, 0), (0, 1), (1, 0), (1, 1)], Color: gray,
 Polyomino: [(2, 0), (2, 1)], Color: gray]
sage: next(it)
[Polyomino: [(1, 0), (1, 1), (2, 0), (2, 1)], Color: gray,
 Polyomino: [(0, 0), (0, 1)], Color: gray]
sage: next(it)
Traceback (most recent call last):
...
StopIteration
sage: T.number_of_solutions()
2

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import TilingSolver, Polyomino
>>> p = Polyomino([(Integer(0),Integer(0)), (Integer(0),Integer(1))])
>>> q = Polyomino([(Integer(0),Integer(0)), (Integer(0),Integer(1)), (Integer(1),Integer(0)), (Integer(1),Integer(1))])
>>> T = TilingSolver([p,q], box=[Integer(3),Integer(2)])
>>> it = T.solve()
>>> next(it)
[Polyomino: [(0, 0), (0, 1), (1, 0), (1, 1)], Color: gray,
 Polyomino: [(2, 0), (2, 1)], Color: gray]
>>> next(it)
[Polyomino: [(1, 0), (1, 1), (2, 0), (2, 1)], Color: gray,
 Polyomino: [(0, 0), (0, 1)], Color: gray]
>>> next(it)
Traceback (most recent call last):
...
StopIteration
>>> T.number_of_solutions()
2

Scott’s pentamino problem

As mentioned in the introduction of [Knuth1], Scott’s pentamino problem consists in tiling a chessboard leaving the center four squares vacant with the 12 distinct pentaminoes.

The 12 pentaminoes:

Sage

sage: from sage.combinat.tiling import Polyomino
sage: I = Polyomino([(0,0),(1,0),(2,0),(3,0),(4,0)], color='brown')
sage: N = Polyomino([(1,0),(1,1),(1,2),(0,2),(0,3)], color='yellow')
sage: L = Polyomino([(0,0),(1,0),(0,1),(0,2),(0,3)], color='magenta')
sage: U = Polyomino([(0,0),(1,0),(0,1),(0,2),(1,2)], color='violet')
sage: X = Polyomino([(1,0),(0,1),(1,1),(1,2),(2,1)], color='pink')
sage: W = Polyomino([(2,0),(2,1),(1,1),(1,2),(0,2)], color='green')
sage: P = Polyomino([(1,0),(2,0),(0,1),(1,1),(2,1)], color='orange')
sage: F = Polyomino([(1,0),(1,1),(0,1),(2,1),(2,2)], color='gray')
sage: Z = Polyomino([(0,0),(1,0),(1,1),(1,2),(2,2)], color='yellow')
sage: T = Polyomino([(0,0),(0,1),(1,1),(2,1),(0,2)], color='red')
sage: Y = Polyomino([(0,0),(1,0),(2,0),(3,0),(2,1)], color='green')
sage: V = Polyomino([(0,0),(0,1),(0,2),(1,0),(2,0)], color='blue')

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import Polyomino
>>> I = Polyomino([(Integer(0),Integer(0)),(Integer(1),Integer(0)),(Integer(2),Integer(0)),(Integer(3),Integer(0)),(Integer(4),Integer(0))], color='brown')
>>> N = Polyomino([(Integer(1),Integer(0)),(Integer(1),Integer(1)),(Integer(1),Integer(2)),(Integer(0),Integer(2)),(Integer(0),Integer(3))], color='yellow')
>>> L = Polyomino([(Integer(0),Integer(0)),(Integer(1),Integer(0)),(Integer(0),Integer(1)),(Integer(0),Integer(2)),(Integer(0),Integer(3))], color='magenta')
>>> U = Polyomino([(Integer(0),Integer(0)),(Integer(1),Integer(0)),(Integer(0),Integer(1)),(Integer(0),Integer(2)),(Integer(1),Integer(2))], color='violet')
>>> X = Polyomino([(Integer(1),Integer(0)),(Integer(0),Integer(1)),(Integer(1),Integer(1)),(Integer(1),Integer(2)),(Integer(2),Integer(1))], color='pink')
>>> W = Polyomino([(Integer(2),Integer(0)),(Integer(2),Integer(1)),(Integer(1),Integer(1)),(Integer(1),Integer(2)),(Integer(0),Integer(2))], color='green')
>>> P = Polyomino([(Integer(1),Integer(0)),(Integer(2),Integer(0)),(Integer(0),Integer(1)),(Integer(1),Integer(1)),(Integer(2),Integer(1))], color='orange')
>>> F = Polyomino([(Integer(1),Integer(0)),(Integer(1),Integer(1)),(Integer(0),Integer(1)),(Integer(2),Integer(1)),(Integer(2),Integer(2))], color='gray')
>>> Z = Polyomino([(Integer(0),Integer(0)),(Integer(1),Integer(0)),(Integer(1),Integer(1)),(Integer(1),Integer(2)),(Integer(2),Integer(2))], color='yellow')
>>> T = Polyomino([(Integer(0),Integer(0)),(Integer(0),Integer(1)),(Integer(1),Integer(1)),(Integer(2),Integer(1)),(Integer(0),Integer(2))], color='red')
>>> Y = Polyomino([(Integer(0),Integer(0)),(Integer(1),Integer(0)),(Integer(2),Integer(0)),(Integer(3),Integer(0)),(Integer(2),Integer(1))], color='green')
>>> V = Polyomino([(Integer(0),Integer(0)),(Integer(0),Integer(1)),(Integer(0),Integer(2)),(Integer(1),Integer(0)),(Integer(2),Integer(0))], color='blue')

A \(8 \times 8\) chessboard leaving the center four squares vacant:

Sage

sage: import itertools
sage: s = set(itertools.product(range(8), repeat=2))
sage: s.difference_update([(3,3), (3,4), (4,3), (4,4)])
sage: chessboard = Polyomino(s)
sage: len(chessboard)
60

Python

>>> from sage.all import *
>>> import itertools
>>> s = set(itertools.product(range(Integer(8)), repeat=Integer(2)))
>>> s.difference_update([(Integer(3),Integer(3)), (Integer(3),Integer(4)), (Integer(4),Integer(3)), (Integer(4),Integer(4))])
>>> chessboard = Polyomino(s)
>>> len(chessboard)
60

This problem is represented by a matrix made of 1568 rows and 72 columns. It has 65 different solutions up to isometries:

Sage

sage: from sage.combinat.tiling import TilingSolver
sage: T = TilingSolver([I,N,L,U,X,W,P,F,Z,T,Y,V], box=chessboard, reflection=True)
sage: T
Tiling solver of 12 pieces into a box of size 60
Rotation allowed: True
Reflection allowed: True
Reusing pieces allowed: False
sage: len(T.rows())                # long time
1568
sage: T.number_of_solutions()      # long time
520
sage: 520 / 8
65

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import TilingSolver
>>> T = TilingSolver([I,N,L,U,X,W,P,F,Z,T,Y,V], box=chessboard, reflection=True)
>>> T
Tiling solver of 12 pieces into a box of size 60
Rotation allowed: True
Reflection allowed: True
Reusing pieces allowed: False
>>> len(T.rows())                # long time
1568
>>> T.number_of_solutions()      # long time
520
>>> Integer(520) / Integer(8)
65

Showing one solution:

Sage

sage: solution = next(T.solve())                                    # long time
sage: G = sum([piece.show2d() for piece in solution], Graphics())   # long time, needs sage.plot
sage: G.show(aspect_ratio=1, axes=False)                            # long time, needs sage.plot

Python

>>> from sage.all import *
>>> solution = next(T.solve())                                    # long time
>>> G = sum([piece.show2d() for piece in solution], Graphics())   # long time, needs sage.plot
>>> G.show(aspect_ratio=Integer(1), axes=False)                            # long time, needs sage.plot

1d Easy Example

Here is an easy one dimensional example where we try to tile a stick of length 6 with three sticks of length 1, 2 and 3. There are six solutions:

Sage

sage: p = Polyomino([[0]])
sage: q = Polyomino([[0],[1]])
sage: r = Polyomino([[0],[1],[2]])
sage: T = TilingSolver([p,q,r], box=[6])
sage: len(T.rows())
15
sage: it = T.solve()
sage: next(it)
[Polyomino: [(0)], Color: gray,
 Polyomino: [(1), (2)], Color: gray,
 Polyomino: [(3), (4), (5)], Color: gray]
sage: next(it)
[Polyomino: [(0)], Color: gray,
 Polyomino: [(1), (2), (3)], Color: gray,
 Polyomino: [(4), (5)], Color: gray]
sage: T.number_of_solutions()
6

Python

>>> from sage.all import *
>>> p = Polyomino([[Integer(0)]])
>>> q = Polyomino([[Integer(0)],[Integer(1)]])
>>> r = Polyomino([[Integer(0)],[Integer(1)],[Integer(2)]])
>>> T = TilingSolver([p,q,r], box=[Integer(6)])
>>> len(T.rows())
15
>>> it = T.solve()
>>> next(it)
[Polyomino: [(0)], Color: gray,
 Polyomino: [(1), (2)], Color: gray,
 Polyomino: [(3), (4), (5)], Color: gray]
>>> next(it)
[Polyomino: [(0)], Color: gray,
 Polyomino: [(1), (2), (3)], Color: gray,
 Polyomino: [(4), (5)], Color: gray]
>>> T.number_of_solutions()
6

2d Puzzle allowing reflections

The following is a puzzle owned by Florent Hivert:

Sage

sage: from sage.combinat.tiling import Polyomino, TilingSolver
sage: L = []
sage: L.append(Polyomino([(0,0),(0,1),(0,2),(0,3),(1,0),(1,1),(1,2),(1,3)], 'yellow'))
sage: L.append(Polyomino([(0,0),(0,1),(0,2),(0,3),(1,0),(1,1),(1,2)], "black"))
sage: L.append(Polyomino([(0,0),(0,1),(0,2),(0,3),(1,0),(1,1),(1,3)], "gray"))
sage: L.append(Polyomino([(0,0),(0,1),(0,2),(0,3),(1,0),(1,3)],"cyan"))
sage: L.append(Polyomino([(0,0),(0,1),(0,2),(0,3),(1,0),(1,1)],"red"))
sage: L.append(Polyomino([(0,0),(0,1),(0,2),(0,3),(1,1),(1,2)],"blue"))
sage: L.append(Polyomino([(0,0),(0,1),(0,2),(0,3),(1,1),(1,3)],"green"))
sage: L.append(Polyomino([(0,1),(0,2),(0,3),(1,0),(1,1),(1,3)],"magenta"))
sage: L.append(Polyomino([(0,1),(0,2),(0,3),(1,0),(1,1),(1,2)],"orange"))
sage: L.append(Polyomino([(0,0),(0,1),(0,2),(1,0),(1,1),(1,2)],"pink"))

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import Polyomino, TilingSolver
>>> L = []
>>> L.append(Polyomino([(Integer(0),Integer(0)),(Integer(0),Integer(1)),(Integer(0),Integer(2)),(Integer(0),Integer(3)),(Integer(1),Integer(0)),(Integer(1),Integer(1)),(Integer(1),Integer(2)),(Integer(1),Integer(3))], 'yellow'))
>>> L.append(Polyomino([(Integer(0),Integer(0)),(Integer(0),Integer(1)),(Integer(0),Integer(2)),(Integer(0),Integer(3)),(Integer(1),Integer(0)),(Integer(1),Integer(1)),(Integer(1),Integer(2))], "black"))
>>> L.append(Polyomino([(Integer(0),Integer(0)),(Integer(0),Integer(1)),(Integer(0),Integer(2)),(Integer(0),Integer(3)),(Integer(1),Integer(0)),(Integer(1),Integer(1)),(Integer(1),Integer(3))], "gray"))
>>> L.append(Polyomino([(Integer(0),Integer(0)),(Integer(0),Integer(1)),(Integer(0),Integer(2)),(Integer(0),Integer(3)),(Integer(1),Integer(0)),(Integer(1),Integer(3))],"cyan"))
>>> L.append(Polyomino([(Integer(0),Integer(0)),(Integer(0),Integer(1)),(Integer(0),Integer(2)),(Integer(0),Integer(3)),(Integer(1),Integer(0)),(Integer(1),Integer(1))],"red"))
>>> L.append(Polyomino([(Integer(0),Integer(0)),(Integer(0),Integer(1)),(Integer(0),Integer(2)),(Integer(0),Integer(3)),(Integer(1),Integer(1)),(Integer(1),Integer(2))],"blue"))
>>> L.append(Polyomino([(Integer(0),Integer(0)),(Integer(0),Integer(1)),(Integer(0),Integer(2)),(Integer(0),Integer(3)),(Integer(1),Integer(1)),(Integer(1),Integer(3))],"green"))
>>> L.append(Polyomino([(Integer(0),Integer(1)),(Integer(0),Integer(2)),(Integer(0),Integer(3)),(Integer(1),Integer(0)),(Integer(1),Integer(1)),(Integer(1),Integer(3))],"magenta"))
>>> L.append(Polyomino([(Integer(0),Integer(1)),(Integer(0),Integer(2)),(Integer(0),Integer(3)),(Integer(1),Integer(0)),(Integer(1),Integer(1)),(Integer(1),Integer(2))],"orange"))
>>> L.append(Polyomino([(Integer(0),Integer(0)),(Integer(0),Integer(1)),(Integer(0),Integer(2)),(Integer(1),Integer(0)),(Integer(1),Integer(1)),(Integer(1),Integer(2))],"pink"))

By default, rotations are allowed and reflections are not. In this case, there are no solutions for tiling a \(8 \times 8\) rectangular box:

Sage

sage: T = TilingSolver(L, box=(8,8))
sage: T.number_of_solutions()                                       # long time (2.5s)
0

Python

>>> from sage.all import *
>>> T = TilingSolver(L, box=(Integer(8),Integer(8)))
>>> T.number_of_solutions()                                       # long time (2.5s)
0

If reflections are allowed, there are solutions. Solve the puzzle and show one solution:

Sage

sage: T = TilingSolver(L, box=(8,8), reflection=True)
sage: solution = next(T.solve())                                    # long time (7s)
sage: G = sum([piece.show2d() for piece in solution], Graphics())   # long time (<1s), needs sage.plot
sage: G.show(aspect_ratio=1, axes=False)                            # long time (2s), needs sage.plot

Python

>>> from sage.all import *
>>> T = TilingSolver(L, box=(Integer(8),Integer(8)), reflection=True)
>>> solution = next(T.solve())                                    # long time (7s)
>>> G = sum([piece.show2d() for piece in solution], Graphics())   # long time (<1s), needs sage.plot
>>> G.show(aspect_ratio=Integer(1), axes=False)                            # long time (2s), needs sage.plot

Compute the number of solutions:

Sage

sage: T.number_of_solutions()                                       # long time (2.6s)
328

Python

>>> from sage.all import *
>>> T.number_of_solutions()                                       # long time (2.6s)
328

Create a animation of all the solutions:

Sage

sage: a = T.animate()                            # not tested
sage: a                                          # not tested
Animation with 328 frames

Python

>>> from sage.all import *
>>> a = T.animate()                            # not tested
>>> a                                          # not tested
Animation with 328 frames

3d Puzzle

The same thing done in 3d without allowing reflections this time:

Sage

sage: from sage.combinat.tiling import Polyomino, TilingSolver
sage: L = []
sage: L.append(Polyomino([(0,0,0),(0,1,0),(0,2,0),(0,3,0),(1,0,0),(1,1,0),(1,2,0),(1,3,0)]))
sage: L.append(Polyomino([(0,0,0),(0,1,0),(0,2,0),(0,3,0),(1,0,0),(1,1,0),(1,2,0)]))
sage: L.append(Polyomino([(0,0,0),(0,1,0),(0,2,0),(0,3,0),(1,0,0),(1,1,0),(1,3,0)]))
sage: L.append(Polyomino([(0,0,0),(0,1,0),(0,2,0),(0,3,0),(1,0,0),(1,3,0)]))
sage: L.append(Polyomino([(0,0,0),(0,1,0),(0,2,0),(0,3,0),(1,0,0),(1,1,0)]))
sage: L.append(Polyomino([(0,0,0),(0,1,0),(0,2,0),(0,3,0),(1,1,0),(1,2,0)]))
sage: L.append(Polyomino([(0,0,0),(0,1,0),(0,2,0),(0,3,0),(1,1,0),(1,3,0)]))
sage: L.append(Polyomino([(0,1,0),(0,2,0),(0,3,0),(1,0,0),(1,1,0),(1,3,0)]))
sage: L.append(Polyomino([(0,1,0),(0,2,0),(0,3,0),(1,0,0),(1,1,0),(1,2,0)]))
sage: L.append(Polyomino([(0,0,0),(0,1,0),(0,2,0),(1,0,0),(1,1,0),(1,2,0)]))

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import Polyomino, TilingSolver
>>> L = []
>>> L.append(Polyomino([(Integer(0),Integer(0),Integer(0)),(Integer(0),Integer(1),Integer(0)),(Integer(0),Integer(2),Integer(0)),(Integer(0),Integer(3),Integer(0)),(Integer(1),Integer(0),Integer(0)),(Integer(1),Integer(1),Integer(0)),(Integer(1),Integer(2),Integer(0)),(Integer(1),Integer(3),Integer(0))]))
>>> L.append(Polyomino([(Integer(0),Integer(0),Integer(0)),(Integer(0),Integer(1),Integer(0)),(Integer(0),Integer(2),Integer(0)),(Integer(0),Integer(3),Integer(0)),(Integer(1),Integer(0),Integer(0)),(Integer(1),Integer(1),Integer(0)),(Integer(1),Integer(2),Integer(0))]))
>>> L.append(Polyomino([(Integer(0),Integer(0),Integer(0)),(Integer(0),Integer(1),Integer(0)),(Integer(0),Integer(2),Integer(0)),(Integer(0),Integer(3),Integer(0)),(Integer(1),Integer(0),Integer(0)),(Integer(1),Integer(1),Integer(0)),(Integer(1),Integer(3),Integer(0))]))
>>> L.append(Polyomino([(Integer(0),Integer(0),Integer(0)),(Integer(0),Integer(1),Integer(0)),(Integer(0),Integer(2),Integer(0)),(Integer(0),Integer(3),Integer(0)),(Integer(1),Integer(0),Integer(0)),(Integer(1),Integer(3),Integer(0))]))
>>> L.append(Polyomino([(Integer(0),Integer(0),Integer(0)),(Integer(0),Integer(1),Integer(0)),(Integer(0),Integer(2),Integer(0)),(Integer(0),Integer(3),Integer(0)),(Integer(1),Integer(0),Integer(0)),(Integer(1),Integer(1),Integer(0))]))
>>> L.append(Polyomino([(Integer(0),Integer(0),Integer(0)),(Integer(0),Integer(1),Integer(0)),(Integer(0),Integer(2),Integer(0)),(Integer(0),Integer(3),Integer(0)),(Integer(1),Integer(1),Integer(0)),(Integer(1),Integer(2),Integer(0))]))
>>> L.append(Polyomino([(Integer(0),Integer(0),Integer(0)),(Integer(0),Integer(1),Integer(0)),(Integer(0),Integer(2),Integer(0)),(Integer(0),Integer(3),Integer(0)),(Integer(1),Integer(1),Integer(0)),(Integer(1),Integer(3),Integer(0))]))
>>> L.append(Polyomino([(Integer(0),Integer(1),Integer(0)),(Integer(0),Integer(2),Integer(0)),(Integer(0),Integer(3),Integer(0)),(Integer(1),Integer(0),Integer(0)),(Integer(1),Integer(1),Integer(0)),(Integer(1),Integer(3),Integer(0))]))
>>> L.append(Polyomino([(Integer(0),Integer(1),Integer(0)),(Integer(0),Integer(2),Integer(0)),(Integer(0),Integer(3),Integer(0)),(Integer(1),Integer(0),Integer(0)),(Integer(1),Integer(1),Integer(0)),(Integer(1),Integer(2),Integer(0))]))
>>> L.append(Polyomino([(Integer(0),Integer(0),Integer(0)),(Integer(0),Integer(1),Integer(0)),(Integer(0),Integer(2),Integer(0)),(Integer(1),Integer(0),Integer(0)),(Integer(1),Integer(1),Integer(0)),(Integer(1),Integer(2),Integer(0))]))

Solve the puzzle and show one solution:

Sage

sage: T = TilingSolver(L, box=(8,8,1))
sage: solution = next(T.solve())                                    # long time (8s)
sage: G = sum([p.show3d(size=0.85) for p in solution], Graphics())  # long time (<1s), needs sage.plot
sage: G.show(aspect_ratio=1, viewer='tachyon')                      # long time (2s), needs sage.plot

Python

>>> from sage.all import *
>>> T = TilingSolver(L, box=(Integer(8),Integer(8),Integer(1)))
>>> solution = next(T.solve())                                    # long time (8s)
>>> G = sum([p.show3d(size=RealNumber('0.85')) for p in solution], Graphics())  # long time (<1s), needs sage.plot
>>> G.show(aspect_ratio=Integer(1), viewer='tachyon')                      # long time (2s), needs sage.plot

Let us compute the number of solutions:

Sage

sage: T.number_of_solutions()                                       # long time (3s)
328

Python

>>> from sage.all import *
>>> T.number_of_solutions()                                       # long time (3s)
328

Donald Knuth example : the Y pentamino

Donald Knuth [Knuth1] considered the problem of packing 45 Y pentaminoes into a \(15 \times 15\) square:

Sage

sage: from sage.combinat.tiling import Polyomino, TilingSolver
sage: y = Polyomino([(0,0),(1,0),(2,0),(3,0),(2,1)])
sage: T = TilingSolver([y], box=(5,10), reusable=True, reflection=True)
sage: T.number_of_solutions()
10
sage: solution = next(T.solve())
sage: G = sum([p.show2d() for p in solution], Graphics())                           # needs sage.plot
sage: G.show(aspect_ratio=1)                                        # long time (2s), needs sage.plot

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import Polyomino, TilingSolver
>>> y = Polyomino([(Integer(0),Integer(0)),(Integer(1),Integer(0)),(Integer(2),Integer(0)),(Integer(3),Integer(0)),(Integer(2),Integer(1))])
>>> T = TilingSolver([y], box=(Integer(5),Integer(10)), reusable=True, reflection=True)
>>> T.number_of_solutions()
10
>>> solution = next(T.solve())
>>> G = sum([p.show2d() for p in solution], Graphics())                           # needs sage.plot
>>> G.show(aspect_ratio=Integer(1))                                        # long time (2s), needs sage.plot

Sage

sage: T = TilingSolver([y], box=(15,15), reusable=True, reflection=True)
sage: T.number_of_solutions()                      # not tested
1696

Python

>>> from sage.all import *
>>> T = TilingSolver([y], box=(Integer(15),Integer(15)), reusable=True, reflection=True)
>>> T.number_of_solutions()                      # not tested
1696

Up to the symmetries of the square, there are 212 distinct solutions:

Sage

sage: 1696 // 8
212

Python

>>> from sage.all import *
>>> Integer(1696) // Integer(8)
212

Animation of Donald Knuth’s dancing links

Animation of the solutions:

Sage

sage: from sage.combinat.tiling import Polyomino, TilingSolver
sage: Y = Polyomino([(0,0),(1,0),(2,0),(3,0),(2,1)], color='yellow')
sage: T = TilingSolver([Y], box=(15,15), reusable=True, reflection=True)
sage: a = T.animate(stop=40); a                     # long time, optional - imagemagick, needs sage.plot
Animation with 40 frames

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import Polyomino, TilingSolver
>>> Y = Polyomino([(Integer(0),Integer(0)),(Integer(1),Integer(0)),(Integer(2),Integer(0)),(Integer(3),Integer(0)),(Integer(2),Integer(1))], color='yellow')
>>> T = TilingSolver([Y], box=(Integer(15),Integer(15)), reusable=True, reflection=True)
>>> a = T.animate(stop=Integer(40)); a                     # long time, optional - imagemagick, needs sage.plot
Animation with 40 frames

Incremental animation of the solutions (one piece is removed/added at a time):

Sage

sage: a = T.animate('incremental', stop=40); a      # long time, optional - imagemagick, needs sage.plot
Animation with 40 frames
sage: a.show(delay=50, iterations=1)                # long time, optional - imagemagick, needs sage.plot

Python

>>> from sage.all import *
>>> a = T.animate('incremental', stop=Integer(40)); a      # long time, optional - imagemagick, needs sage.plot
Animation with 40 frames
>>> a.show(delay=Integer(50), iterations=Integer(1))                # long time, optional - imagemagick, needs sage.plot

5d Easy Example

Here is a 5d example. Let us try to fill the \(2 \times 2 \times 2 \times 2 \times 2\) rectangle with reusable \(1 \times 1 \times 1 \times 1 \times 1\) rectangles. Obviously, there is one solution:

Sage

sage: from sage.combinat.tiling import Polyomino, TilingSolver
sage: p = Polyomino([(0,0,0,0,0)])
sage: T = TilingSolver([p], box=(2,2,2,2,2), reusable=True)
sage: rows = T.rows()                               # long time (3s)
sage: rows                                          # long time (fast)
[[0], [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12],
 [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24],
 [25], [26], [27], [28], [29], [30], [31]]
sage: T.number_of_solutions()                       # long time (fast)
1

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import Polyomino, TilingSolver
>>> p = Polyomino([(Integer(0),Integer(0),Integer(0),Integer(0),Integer(0))])
>>> T = TilingSolver([p], box=(Integer(2),Integer(2),Integer(2),Integer(2),Integer(2)), reusable=True)
>>> rows = T.rows()                               # long time (3s)
>>> rows                                          # long time (fast)
[[0], [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12],
 [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24],
 [25], [26], [27], [28], [29], [30], [31]]
>>> T.number_of_solutions()                       # long time (fast)
1

REFERENCES:

[Knuth1] (1,2,3)

Knuth, Donald (2000). “Dancing links”. arXiv cs/0011047.

class sage.combinat.tiling.Polyomino(coords, color='gray', dimension=None)

Bases: SageObject

A polyomino in \(\ZZ^d\).

The polyomino is the union of the unit square (or cube, or n-cube) centered at those coordinates. Such an object should be connected, but the code does not make this assumption.

INPUT:

• coords – iterable of integer coordinates in \(\ZZ^d\)

• color – string (default: 'gray'); color for display

• dimension – integer (default: None); dimension of the space, if None, it is guessed from the coords if coords is non empty

EXAMPLES:

Sage

sage: from sage.combinat.tiling import Polyomino
sage: Polyomino([(0,0,0), (0,1,0), (1,1,0), (1,1,1)], color='blue')
Polyomino: [(0, 0, 0), (0, 1, 0), (1, 1, 0), (1, 1, 1)], Color: blue

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import Polyomino
>>> Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(1),Integer(0)), (Integer(1),Integer(1),Integer(0)), (Integer(1),Integer(1),Integer(1))], color='blue')
Polyomino: [(0, 0, 0), (0, 1, 0), (1, 1, 0), (1, 1, 1)], Color: blue

boundary()

Return the boundary of a 2d polyomino.

INPUT:

• self – a 2d polyomino

OUTPUT:

• list of edges (an edge is a pair of adjacent 2d coordinates)

EXAMPLES:

Sage

sage: from sage.combinat.tiling import Polyomino
sage: p = Polyomino([(0,0), (1,0), (0,1), (1,1)])
sage: sorted(p.boundary())
[((-0.5, -0.5), (-0.5, 0.5)), ((-0.5, -0.5), (0.5, -0.5)),
 ((-0.5, 0.5), (-0.5, 1.5)), ((-0.5, 1.5), (0.5, 1.5)),
 ((0.5, -0.5), (1.5, -0.5)), ((0.5, 1.5), (1.5, 1.5)),
 ((1.5, -0.5), (1.5, 0.5)), ((1.5, 0.5), (1.5, 1.5))]
sage: len(_)
8
sage: p = Polyomino([(5,5)])
sage: sorted(p.boundary())
[((4.5, 4.5), (4.5, 5.5)), ((4.5, 4.5), (5.5, 4.5)),
 ((4.5, 5.5), (5.5, 5.5)), ((5.5, 4.5), (5.5, 5.5))]

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import Polyomino
>>> p = Polyomino([(Integer(0),Integer(0)), (Integer(1),Integer(0)), (Integer(0),Integer(1)), (Integer(1),Integer(1))])
>>> sorted(p.boundary())
[((-0.5, -0.5), (-0.5, 0.5)), ((-0.5, -0.5), (0.5, -0.5)),
 ((-0.5, 0.5), (-0.5, 1.5)), ((-0.5, 1.5), (0.5, 1.5)),
 ((0.5, -0.5), (1.5, -0.5)), ((0.5, 1.5), (1.5, 1.5)),
 ((1.5, -0.5), (1.5, 0.5)), ((1.5, 0.5), (1.5, 1.5))]
>>> len(_)
8
>>> p = Polyomino([(Integer(5),Integer(5))])
>>> sorted(p.boundary())
[((4.5, 4.5), (4.5, 5.5)), ((4.5, 4.5), (5.5, 4.5)),
 ((4.5, 5.5), (5.5, 5.5)), ((5.5, 4.5), (5.5, 5.5))]

bounding_box()

EXAMPLES:

Sage

sage: from sage.combinat.tiling import Polyomino
sage: p = Polyomino([(0,0,0),(1,0,0),(1,1,0),(1,1,1),(1,2,0)], color='deeppink')
sage: p.bounding_box()
[[0, 0, 0], [1, 2, 1]]

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import Polyomino
>>> p = Polyomino([(Integer(0),Integer(0),Integer(0)),(Integer(1),Integer(0),Integer(0)),(Integer(1),Integer(1),Integer(0)),(Integer(1),Integer(1),Integer(1)),(Integer(1),Integer(2),Integer(0))], color='deeppink')
>>> p.bounding_box()
[[0, 0, 0], [1, 2, 1]]

canonical()

Return the translated copy of self having minimal and nonnegative coordinates

EXAMPLES:

Sage

sage: from sage.combinat.tiling import Polyomino
sage: p = Polyomino([(0,0,0),(1,0,0),(1,1,0),(1,1,1),(1,2,0)], color='deeppink')
sage: p
Polyomino: [(0, 0, 0), (1, 0, 0), (1, 1, 0), (1, 1, 1), (1, 2, 0)], Color: deeppink
sage: p.canonical()
Polyomino: [(0, 0, 0), (1, 0, 0), (1, 1, 0), (1, 1, 1), (1, 2, 0)], Color: deeppink

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import Polyomino
>>> p = Polyomino([(Integer(0),Integer(0),Integer(0)),(Integer(1),Integer(0),Integer(0)),(Integer(1),Integer(1),Integer(0)),(Integer(1),Integer(1),Integer(1)),(Integer(1),Integer(2),Integer(0))], color='deeppink')
>>> p
Polyomino: [(0, 0, 0), (1, 0, 0), (1, 1, 0), (1, 1, 1), (1, 2, 0)], Color: deeppink
>>> p.canonical()
Polyomino: [(0, 0, 0), (1, 0, 0), (1, 1, 0), (1, 1, 1), (1, 2, 0)], Color: deeppink

canonical_isometric_copies(orientation_preserving=True, mod_box_isometries=False)

Return the list of image of self under isometries of the \(n\)-cube where the coordinates are all nonnegative and minimal.

INPUT:

• orientation_preserving – boolean (default: True); if True, the group of isometries of the \(n\)-cube is restricted to those that preserve the orientation, i.e. of determinant 1.

• mod_box_isometries – boolean (default: False); whether to quotient the group of isometries of the \(n\)-cube by the subgroup of isometries of the \(a_1\times a_2\cdots \times a_n\) rectangular box where are the \(a_i\) are assumed to be distinct.

OUTPUT: set of Polyomino

EXAMPLES:

Sage

sage: from sage.combinat.tiling import Polyomino
sage: p = Polyomino([(0,0,0), (0,1,0), (1,1,0), (1,1,1)], color='blue')
sage: s = p.canonical_isometric_copies()
sage: len(s)
12

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import Polyomino
>>> p = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(1),Integer(0)), (Integer(1),Integer(1),Integer(0)), (Integer(1),Integer(1),Integer(1))], color='blue')
>>> s = p.canonical_isometric_copies()
>>> len(s)
12

With the non orientation-preserving:

Sage

sage: s = p.canonical_isometric_copies(orientation_preserving=False)
sage: len(s)
24

Python

>>> from sage.all import *
>>> s = p.canonical_isometric_copies(orientation_preserving=False)
>>> len(s)
24

Modulo rotation by angle 180 degrees:

Sage

sage: s = p.canonical_isometric_copies(mod_box_isometries=True)
sage: len(s)
3

Python

>>> from sage.all import *
>>> s = p.canonical_isometric_copies(mod_box_isometries=True)
>>> len(s)
3

center()

Return the center of the polyomino.

EXAMPLES:

Sage

sage: from sage.combinat.tiling import Polyomino
sage: p = Polyomino([(0,0,0),(0,0,1)])
sage: p.center()
(0, 0, 1/2)

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import Polyomino
>>> p = Polyomino([(Integer(0),Integer(0),Integer(0)),(Integer(0),Integer(0),Integer(1))])
>>> p.center()
(0, 0, 1/2)

In 3d:

Sage

sage: p = Polyomino([(0,0,0),(1,0,0),(1,1,0),(1,1,1),(1,2,0)], color='deeppink')
sage: p.center()
(4/5, 4/5, 1/5)

Python

>>> from sage.all import *
>>> p = Polyomino([(Integer(0),Integer(0),Integer(0)),(Integer(1),Integer(0),Integer(0)),(Integer(1),Integer(1),Integer(0)),(Integer(1),Integer(1),Integer(1)),(Integer(1),Integer(2),Integer(0))], color='deeppink')
>>> p.center()
(4/5, 4/5, 1/5)

In 2d:

Sage

sage: p = Polyomino([(0,0),(1,0),(1,1),(1,2)])
sage: p.center()
(3/4, 3/4)

Python

>>> from sage.all import *
>>> p = Polyomino([(Integer(0),Integer(0)),(Integer(1),Integer(0)),(Integer(1),Integer(1)),(Integer(1),Integer(2))])
>>> p.center()
(3/4, 3/4)

color(color=None)

Return or change the color of the polyomino.

INPUT:

• color – string, RBG tuple or None (default: None), if None, it returns the current color

EXAMPLES:

Sage

sage: from sage.combinat.tiling import Polyomino
sage: p = Polyomino([(0,0,0), (0,1,0), (1,1,0), (1,1,1)], color='blue')
sage: p.color()
'blue'

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import Polyomino
>>> p = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(1),Integer(0)), (Integer(1),Integer(1),Integer(0)), (Integer(1),Integer(1),Integer(1))], color='blue')
>>> p.color()
'blue'

frozenset()

Return the elements of \(\ZZ^d\) in the polyomino as a frozenset.

EXAMPLES:

Sage

sage: from sage.combinat.tiling import Polyomino
sage: p = Polyomino([(0,0,0), (0,1,0), (1,1,0), (1,1,1)], color='red')
sage: p.frozenset()
frozenset({(0, 0, 0), (0, 1, 0), (1, 1, 0), (1, 1, 1)})

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import Polyomino
>>> p = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(1),Integer(0)), (Integer(1),Integer(1),Integer(0)), (Integer(1),Integer(1),Integer(1))], color='red')
>>> p.frozenset()
frozenset({(0, 0, 0), (0, 1, 0), (1, 1, 0), (1, 1, 1)})

intersection(other)

Return the intersection of self and other.

INPUT:

• other – a polyomino

OUTPUT: polyomino

EXAMPLES:

Sage

sage: from sage.combinat.tiling import Polyomino
sage: a = Polyomino([(0,0)])
sage: b = Polyomino([(0,0), (0,1), (1,1), (2,1)])
sage: a.intersection(b)
Polyomino: [(0, 0)], Color: gray
sage: a.intersection(b+(1,1))
Polyomino: [], Color: gray

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import Polyomino
>>> a = Polyomino([(Integer(0),Integer(0))])
>>> b = Polyomino([(Integer(0),Integer(0)), (Integer(0),Integer(1)), (Integer(1),Integer(1)), (Integer(2),Integer(1))])
>>> a.intersection(b)
Polyomino: [(0, 0)], Color: gray
>>> a.intersection(b+(Integer(1),Integer(1)))
Polyomino: [], Color: gray

isometric_copies(box, orientation_preserving=True, mod_box_isometries=False)

Return the translated and isometric images of self that lies in the box.

INPUT:

• box – polyomino or tuple of integers (size of a box)

• orientation_preserving – boolean (default: True); if True, the group of isometries of the \(n\)-cube is restricted to those that preserve the orientation, i.e. of determinant 1.

• mod_box_isometries – boolean (default: False); whether to quotient the group of isometries of the \(n\)-cube by the subgroup of isometries of the \(a_1\times a_2\cdots \times a_n\) rectangular box where are the \(a_i\) are assumed to be distinct.

EXAMPLES:

Sage

sage: from sage.combinat.tiling import Polyomino
sage: p = Polyomino([(0,0,0),(1,0,0),(1,1,0),(1,1,1),(1,2,0)], color='deeppink')
sage: L = list(p.isometric_copies(box=(5,8,2)))
sage: len(L)
360

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import Polyomino
>>> p = Polyomino([(Integer(0),Integer(0),Integer(0)),(Integer(1),Integer(0),Integer(0)),(Integer(1),Integer(1),Integer(0)),(Integer(1),Integer(1),Integer(1)),(Integer(1),Integer(2),Integer(0))], color='deeppink')
>>> L = list(p.isometric_copies(box=(Integer(5),Integer(8),Integer(2))))
>>> len(L)
360

Sage

sage: p = Polyomino([(0,0,0),(1,0,0),(1,1,0),(1,2,0),(1,2,1)], color='orange')
sage: L = list(p.isometric_copies(box=(5,8,2)))
sage: len(L)
180
sage: L = list(p.isometric_copies((5,8,2), False))
sage: len(L)
360
sage: L = list(p.isometric_copies((5,8,2), mod_box_isometries=True))
sage: len(L)
45

Python

>>> from sage.all import *
>>> p = Polyomino([(Integer(0),Integer(0),Integer(0)),(Integer(1),Integer(0),Integer(0)),(Integer(1),Integer(1),Integer(0)),(Integer(1),Integer(2),Integer(0)),(Integer(1),Integer(2),Integer(1))], color='orange')
>>> L = list(p.isometric_copies(box=(Integer(5),Integer(8),Integer(2))))
>>> len(L)
180
>>> L = list(p.isometric_copies((Integer(5),Integer(8),Integer(2)), False))
>>> len(L)
360
>>> L = list(p.isometric_copies((Integer(5),Integer(8),Integer(2)), mod_box_isometries=True))
>>> len(L)
45

Sage

sage: p = Polyomino([(0,0), (1,0), (0,1)])
sage: b = Polyomino([(0,0), (1,0), (2,0), (0,1), (1,1), (0,2)])
sage: sorted(p.isometric_copies(b), key=lambda p: p.sorted_list())
[Polyomino: [(0, 0), (0, 1), (1, 0)], Color: gray,
 Polyomino: [(0, 0), (0, 1), (1, 1)], Color: gray,
 Polyomino: [(0, 0), (1, 0), (1, 1)], Color: gray,
 Polyomino: [(0, 1), (0, 2), (1, 1)], Color: gray,
 Polyomino: [(0, 1), (1, 0), (1, 1)], Color: gray,
 Polyomino: [(1, 0), (1, 1), (2, 0)], Color: gray]

Python

>>> from sage.all import *
>>> p = Polyomino([(Integer(0),Integer(0)), (Integer(1),Integer(0)), (Integer(0),Integer(1))])
>>> b = Polyomino([(Integer(0),Integer(0)), (Integer(1),Integer(0)), (Integer(2),Integer(0)), (Integer(0),Integer(1)), (Integer(1),Integer(1)), (Integer(0),Integer(2))])
>>> sorted(p.isometric_copies(b), key=lambda p: p.sorted_list())
[Polyomino: [(0, 0), (0, 1), (1, 0)], Color: gray,
 Polyomino: [(0, 0), (0, 1), (1, 1)], Color: gray,
 Polyomino: [(0, 0), (1, 0), (1, 1)], Color: gray,
 Polyomino: [(0, 1), (0, 2), (1, 1)], Color: gray,
 Polyomino: [(0, 1), (1, 0), (1, 1)], Color: gray,
 Polyomino: [(1, 0), (1, 1), (2, 0)], Color: gray]

isometric_copies_intersection(box, orientation_preserving=True)

Return the set of non empty intersections of isometric images of self with a polyomino.

INPUT:

• box – polyomino or tuple of integers (size of a box)

• orientation_preserving – boolean (default: True); if True, the group of isometries of the \(n\)-cube is restricted to those that preserve the orientation, i.e. of determinant 1.

EXAMPLES:

Sage

sage: from sage.combinat.tiling import Polyomino
sage: p = Polyomino([(0,0),(1,0)], color='deeppink')
sage: sorted(sorted(a.frozenset()) for a in p.isometric_copies_intersection(box=(2,3)))
[[(0, 0)],
 [(0, 0), (0, 1)],
 [(0, 0), (1, 0)],
 [(0, 1)],
 [(0, 1), (0, 2)],
 [(0, 1), (1, 1)],
 [(0, 2)],
 [(0, 2), (1, 2)],
 [(1, 0)],
 [(1, 0), (1, 1)],
 [(1, 1)],
 [(1, 1), (1, 2)],
 [(1, 2)]]

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import Polyomino
>>> p = Polyomino([(Integer(0),Integer(0)),(Integer(1),Integer(0))], color='deeppink')
>>> sorted(sorted(a.frozenset()) for a in p.isometric_copies_intersection(box=(Integer(2),Integer(3))))
[[(0, 0)],
 [(0, 0), (0, 1)],
 [(0, 0), (1, 0)],
 [(0, 1)],
 [(0, 1), (0, 2)],
 [(0, 1), (1, 1)],
 [(0, 2)],
 [(0, 2), (1, 2)],
 [(1, 0)],
 [(1, 0), (1, 1)],
 [(1, 1)],
 [(1, 1), (1, 2)],
 [(1, 2)]]

neighbor_edges()

Return an iterator over the pairs of neighbor coordinates inside of the polyomino.

Two points \(P\) and \(Q\) in the polyomino are neighbor if \(P - Q\) has one coordinate equal to \(+1\) or \(-1\) and zero everywhere else.

EXAMPLES:

Sage

sage: from sage.combinat.tiling import Polyomino
sage: p = Polyomino([(0,0,0),(0,0,1)])
sage: [sorted(edge) for edge in p.neighbor_edges()]
[[(0, 0, 0), (0, 0, 1)]]

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import Polyomino
>>> p = Polyomino([(Integer(0),Integer(0),Integer(0)),(Integer(0),Integer(0),Integer(1))])
>>> [sorted(edge) for edge in p.neighbor_edges()]
[[(0, 0, 0), (0, 0, 1)]]

In 3d:

Sage

sage: p = Polyomino([(0,0,0),(1,0,0),(1,1,0),(1,1,1),(1,2,0)], color='deeppink')
sage: L = sorted(sorted(edge) for edge in p.neighbor_edges())
sage: for a in L: a
[(0, 0, 0), (1, 0, 0)]
[(1, 0, 0), (1, 1, 0)]
[(1, 1, 0), (1, 1, 1)]
[(1, 1, 0), (1, 2, 0)]

Python

>>> from sage.all import *
>>> p = Polyomino([(Integer(0),Integer(0),Integer(0)),(Integer(1),Integer(0),Integer(0)),(Integer(1),Integer(1),Integer(0)),(Integer(1),Integer(1),Integer(1)),(Integer(1),Integer(2),Integer(0))], color='deeppink')
>>> L = sorted(sorted(edge) for edge in p.neighbor_edges())
>>> for a in L: a
[(0, 0, 0), (1, 0, 0)]
[(1, 0, 0), (1, 1, 0)]
[(1, 1, 0), (1, 1, 1)]
[(1, 1, 0), (1, 2, 0)]

In 2d:

Sage

sage: p = Polyomino([(0,0),(1,0),(1,1),(1,2)])
sage: L = sorted(sorted(edge) for edge in p.neighbor_edges())
sage: for a in L: a
[(0, 0), (1, 0)]
[(1, 0), (1, 1)]
[(1, 1), (1, 2)]

Python

>>> from sage.all import *
>>> p = Polyomino([(Integer(0),Integer(0)),(Integer(1),Integer(0)),(Integer(1),Integer(1)),(Integer(1),Integer(2))])
>>> L = sorted(sorted(edge) for edge in p.neighbor_edges())
>>> for a in L: a
[(0, 0), (1, 0)]
[(1, 0), (1, 1)]
[(1, 1), (1, 2)]

self_surrounding(radius, remove_incomplete_copies=True, ncpus=None)

Return a list of isometric copies of self surrounding it with an annulus of given radius.

INPUT:

• self – a polyomino of dimension 2

• radius – integer

• remove_incomplete_copies – boolean (default: True); whether to keep only complete copies of self in the output

• ncpus – integer (default: None); maximal number of subprocesses to use at the same time. If None, it detects the number of effective CPUs in the system using sage.parallel.ncpus.ncpus(). If ncpus=1, the first solution is searched serially.

OUTPUT: list of polyominoes

EXAMPLES:

Sage

sage: from sage.combinat.tiling import Polyomino
sage: H = Polyomino([(-1, 1), (-1, 4), (-1, 7), (0, 0), (0, 1), (0, 2),
....: (0, 3), (0, 4), (0, 5), (0, 6), (0, 7), (0, 8), (1, 1), (1, 2),
....: (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (1, 8), (2, 0), (2, 2),
....: (2, 3), (2, 5), (2, 6), (2, 8)])
sage: solution = H.self_surrounding(8)
sage: G = sum([p.show2d() for p in solution], Graphics())                   # needs sage.plot

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import Polyomino
>>> H = Polyomino([(-Integer(1), Integer(1)), (-Integer(1), Integer(4)), (-Integer(1), Integer(7)), (Integer(0), Integer(0)), (Integer(0), Integer(1)), (Integer(0), Integer(2)),
... (Integer(0), Integer(3)), (Integer(0), Integer(4)), (Integer(0), Integer(5)), (Integer(0), Integer(6)), (Integer(0), Integer(7)), (Integer(0), Integer(8)), (Integer(1), Integer(1)), (Integer(1), Integer(2)),
... (Integer(1), Integer(3)), (Integer(1), Integer(4)), (Integer(1), Integer(5)), (Integer(1), Integer(6)), (Integer(1), Integer(7)), (Integer(1), Integer(8)), (Integer(2), Integer(0)), (Integer(2), Integer(2)),
... (Integer(2), Integer(3)), (Integer(2), Integer(5)), (Integer(2), Integer(6)), (Integer(2), Integer(8))])
>>> solution = H.self_surrounding(Integer(8))
>>> G = sum([p.show2d() for p in solution], Graphics())                   # needs sage.plot

Sage

sage: solution = H.self_surrounding(8, remove_incomplete_copies=False)
sage: G = sum([p.show2d() for p in solution], Graphics())                   # needs sage.plot

Python

>>> from sage.all import *
>>> solution = H.self_surrounding(Integer(8), remove_incomplete_copies=False)
>>> G = sum([p.show2d() for p in solution], Graphics())                   # needs sage.plot

show2d(size=0.7, color='black', thickness=1)

Return a 2d Graphic object representing the polyomino.

INPUT:

• self – a polyomino of dimension 2

• size – number (default: 0.7); the size of each square

• color – color (default: 'black'); color of the boundary line

• thickness – number (default: 1); how thick the boundary line is

EXAMPLES:

Sage

sage: from sage.combinat.tiling import Polyomino
sage: p = Polyomino([(0,0),(1,0),(1,1),(1,2)], color='deeppink')
sage: p.show2d()                    # long time (0.5s)                      # needs sage.plot
Graphics object consisting of 17 graphics primitives

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import Polyomino
>>> p = Polyomino([(Integer(0),Integer(0)),(Integer(1),Integer(0)),(Integer(1),Integer(1)),(Integer(1),Integer(2))], color='deeppink')
>>> p.show2d()                    # long time (0.5s)                      # needs sage.plot
Graphics object consisting of 17 graphics primitives

show3d(size=1)

Return a 3d Graphic object representing the polyomino.

INPUT:

• self – a polyomino of dimension 3

• size – number (default: 1); the size of each 1 \times 1 \times 1 cube. This does a homothety with respect to the center of the polyomino.

EXAMPLES:

Sage

sage: from sage.combinat.tiling import Polyomino
sage: p = Polyomino([(0,0,0), (0,1,0), (1,1,0), (1,1,1)], color='blue')
sage: p.show3d()                    # long time (2s)                        # needs sage.plot
Graphics3d Object

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import Polyomino
>>> p = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(1),Integer(0)), (Integer(1),Integer(1),Integer(0)), (Integer(1),Integer(1),Integer(1))], color='blue')
>>> p.show3d()                    # long time (2s)                        # needs sage.plot
Graphics3d Object

sorted_list()

Return the color of the polyomino.

EXAMPLES:

Sage

sage: from sage.combinat.tiling import Polyomino
sage: p = Polyomino([(0,0,0), (0,1,0), (1,1,0), (1,1,1)], color='blue')
sage: p.sorted_list()
[(0, 0, 0), (0, 1, 0), (1, 1, 0), (1, 1, 1)]

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import Polyomino
>>> p = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(1),Integer(0)), (Integer(1),Integer(1),Integer(0)), (Integer(1),Integer(1),Integer(1))], color='blue')
>>> p.sorted_list()
[(0, 0, 0), (0, 1, 0), (1, 1, 0), (1, 1, 1)]

translated_copies(box)

Return an iterator over the translated images of self inside a polyomino.

INPUT:

• box – polyomino or tuple of integers (size of a box)

OUTPUT: iterator of 3d polyominoes

EXAMPLES:

Sage

sage: from sage.combinat.tiling import Polyomino
sage: p = Polyomino([(0,0,0),(1,0,0),(1,1,0),(1,1,1),(1,2,0)], color='deeppink')
sage: for t in p.translated_copies(box=(5,8,2)): t
Polyomino: [(0, 0, 0), (1, 0, 0), (1, 1, 0), (1, 1, 1), (1, 2, 0)], Color: deeppink
Polyomino: [(0, 1, 0), (1, 1, 0), (1, 2, 0), (1, 2, 1), (1, 3, 0)], Color: deeppink
Polyomino: [(0, 2, 0), (1, 2, 0), (1, 3, 0), (1, 3, 1), (1, 4, 0)], Color: deeppink
Polyomino: [(0, 3, 0), (1, 3, 0), (1, 4, 0), (1, 4, 1), (1, 5, 0)], Color: deeppink
Polyomino: [(0, 4, 0), (1, 4, 0), (1, 5, 0), (1, 5, 1), (1, 6, 0)], Color: deeppink
Polyomino: [(0, 5, 0), (1, 5, 0), (1, 6, 0), (1, 6, 1), (1, 7, 0)], Color: deeppink
Polyomino: [(1, 0, 0), (2, 0, 0), (2, 1, 0), (2, 1, 1), (2, 2, 0)], Color: deeppink
Polyomino: [(1, 1, 0), (2, 1, 0), (2, 2, 0), (2, 2, 1), (2, 3, 0)], Color: deeppink
Polyomino: [(1, 2, 0), (2, 2, 0), (2, 3, 0), (2, 3, 1), (2, 4, 0)], Color: deeppink
Polyomino: [(1, 3, 0), (2, 3, 0), (2, 4, 0), (2, 4, 1), (2, 5, 0)], Color: deeppink
Polyomino: [(1, 4, 0), (2, 4, 0), (2, 5, 0), (2, 5, 1), (2, 6, 0)], Color: deeppink
Polyomino: [(1, 5, 0), (2, 5, 0), (2, 6, 0), (2, 6, 1), (2, 7, 0)], Color: deeppink
Polyomino: [(2, 0, 0), (3, 0, 0), (3, 1, 0), (3, 1, 1), (3, 2, 0)], Color: deeppink
Polyomino: [(2, 1, 0), (3, 1, 0), (3, 2, 0), (3, 2, 1), (3, 3, 0)], Color: deeppink
Polyomino: [(2, 2, 0), (3, 2, 0), (3, 3, 0), (3, 3, 1), (3, 4, 0)], Color: deeppink
Polyomino: [(2, 3, 0), (3, 3, 0), (3, 4, 0), (3, 4, 1), (3, 5, 0)], Color: deeppink
Polyomino: [(2, 4, 0), (3, 4, 0), (3, 5, 0), (3, 5, 1), (3, 6, 0)], Color: deeppink
Polyomino: [(2, 5, 0), (3, 5, 0), (3, 6, 0), (3, 6, 1), (3, 7, 0)], Color: deeppink
Polyomino: [(3, 0, 0), (4, 0, 0), (4, 1, 0), (4, 1, 1), (4, 2, 0)], Color: deeppink
Polyomino: [(3, 1, 0), (4, 1, 0), (4, 2, 0), (4, 2, 1), (4, 3, 0)], Color: deeppink
Polyomino: [(3, 2, 0), (4, 2, 0), (4, 3, 0), (4, 3, 1), (4, 4, 0)], Color: deeppink
Polyomino: [(3, 3, 0), (4, 3, 0), (4, 4, 0), (4, 4, 1), (4, 5, 0)], Color: deeppink
Polyomino: [(3, 4, 0), (4, 4, 0), (4, 5, 0), (4, 5, 1), (4, 6, 0)], Color: deeppink
Polyomino: [(3, 5, 0), (4, 5, 0), (4, 6, 0), (4, 6, 1), (4, 7, 0)], Color: deeppink

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import Polyomino
>>> p = Polyomino([(Integer(0),Integer(0),Integer(0)),(Integer(1),Integer(0),Integer(0)),(Integer(1),Integer(1),Integer(0)),(Integer(1),Integer(1),Integer(1)),(Integer(1),Integer(2),Integer(0))], color='deeppink')
>>> for t in p.translated_copies(box=(Integer(5),Integer(8),Integer(2))): t
Polyomino: [(0, 0, 0), (1, 0, 0), (1, 1, 0), (1, 1, 1), (1, 2, 0)], Color: deeppink
Polyomino: [(0, 1, 0), (1, 1, 0), (1, 2, 0), (1, 2, 1), (1, 3, 0)], Color: deeppink
Polyomino: [(0, 2, 0), (1, 2, 0), (1, 3, 0), (1, 3, 1), (1, 4, 0)], Color: deeppink
Polyomino: [(0, 3, 0), (1, 3, 0), (1, 4, 0), (1, 4, 1), (1, 5, 0)], Color: deeppink
Polyomino: [(0, 4, 0), (1, 4, 0), (1, 5, 0), (1, 5, 1), (1, 6, 0)], Color: deeppink
Polyomino: [(0, 5, 0), (1, 5, 0), (1, 6, 0), (1, 6, 1), (1, 7, 0)], Color: deeppink
Polyomino: [(1, 0, 0), (2, 0, 0), (2, 1, 0), (2, 1, 1), (2, 2, 0)], Color: deeppink
Polyomino: [(1, 1, 0), (2, 1, 0), (2, 2, 0), (2, 2, 1), (2, 3, 0)], Color: deeppink
Polyomino: [(1, 2, 0), (2, 2, 0), (2, 3, 0), (2, 3, 1), (2, 4, 0)], Color: deeppink
Polyomino: [(1, 3, 0), (2, 3, 0), (2, 4, 0), (2, 4, 1), (2, 5, 0)], Color: deeppink
Polyomino: [(1, 4, 0), (2, 4, 0), (2, 5, 0), (2, 5, 1), (2, 6, 0)], Color: deeppink
Polyomino: [(1, 5, 0), (2, 5, 0), (2, 6, 0), (2, 6, 1), (2, 7, 0)], Color: deeppink
Polyomino: [(2, 0, 0), (3, 0, 0), (3, 1, 0), (3, 1, 1), (3, 2, 0)], Color: deeppink
Polyomino: [(2, 1, 0), (3, 1, 0), (3, 2, 0), (3, 2, 1), (3, 3, 0)], Color: deeppink
Polyomino: [(2, 2, 0), (3, 2, 0), (3, 3, 0), (3, 3, 1), (3, 4, 0)], Color: deeppink
Polyomino: [(2, 3, 0), (3, 3, 0), (3, 4, 0), (3, 4, 1), (3, 5, 0)], Color: deeppink
Polyomino: [(2, 4, 0), (3, 4, 0), (3, 5, 0), (3, 5, 1), (3, 6, 0)], Color: deeppink
Polyomino: [(2, 5, 0), (3, 5, 0), (3, 6, 0), (3, 6, 1), (3, 7, 0)], Color: deeppink
Polyomino: [(3, 0, 0), (4, 0, 0), (4, 1, 0), (4, 1, 1), (4, 2, 0)], Color: deeppink
Polyomino: [(3, 1, 0), (4, 1, 0), (4, 2, 0), (4, 2, 1), (4, 3, 0)], Color: deeppink
Polyomino: [(3, 2, 0), (4, 2, 0), (4, 3, 0), (4, 3, 1), (4, 4, 0)], Color: deeppink
Polyomino: [(3, 3, 0), (4, 3, 0), (4, 4, 0), (4, 4, 1), (4, 5, 0)], Color: deeppink
Polyomino: [(3, 4, 0), (4, 4, 0), (4, 5, 0), (4, 5, 1), (4, 6, 0)], Color: deeppink
Polyomino: [(3, 5, 0), (4, 5, 0), (4, 6, 0), (4, 6, 1), (4, 7, 0)], Color: deeppink

This method is independent of the translation of the polyomino:

Sage

sage: q = Polyomino([(0,0,0), (1,0,0)])
sage: list(q.translated_copies((2,2,1)))
[Polyomino: [(0, 0, 0), (1, 0, 0)], Color: gray,
 Polyomino: [(0, 1, 0), (1, 1, 0)], Color: gray]
sage: q = Polyomino([(34,7,-9), (35,7,-9)])
sage: list(q.translated_copies((2,2,1)))
[Polyomino: [(0, 0, 0), (1, 0, 0)], Color: gray,
 Polyomino: [(0, 1, 0), (1, 1, 0)], Color: gray]

Python

>>> from sage.all import *
>>> q = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(1),Integer(0),Integer(0))])
>>> list(q.translated_copies((Integer(2),Integer(2),Integer(1))))
[Polyomino: [(0, 0, 0), (1, 0, 0)], Color: gray,
 Polyomino: [(0, 1, 0), (1, 1, 0)], Color: gray]
>>> q = Polyomino([(Integer(34),Integer(7),-Integer(9)), (Integer(35),Integer(7),-Integer(9))])
>>> list(q.translated_copies((Integer(2),Integer(2),Integer(1))))
[Polyomino: [(0, 0, 0), (1, 0, 0)], Color: gray,
 Polyomino: [(0, 1, 0), (1, 1, 0)], Color: gray]

Inside smaller boxes:

Sage

sage: list(p.translated_copies(box=(2,2,3)))
[]
sage: list(p.translated_copies(box=(2,3,2)))
[Polyomino: [(0, 0, 0), (1, 0, 0), (1, 1, 0), (1, 1, 1), (1, 2, 0)], Color: deeppink]
sage: list(p.translated_copies(box=(3,2,2)))
[]
sage: list(p.translated_copies(box=(1,1,1)))
[]

Python

>>> from sage.all import *
>>> list(p.translated_copies(box=(Integer(2),Integer(2),Integer(3))))
[]
>>> list(p.translated_copies(box=(Integer(2),Integer(3),Integer(2))))
[Polyomino: [(0, 0, 0), (1, 0, 0), (1, 1, 0), (1, 1, 1), (1, 2, 0)], Color: deeppink]
>>> list(p.translated_copies(box=(Integer(3),Integer(2),Integer(2))))
[]
>>> list(p.translated_copies(box=(Integer(1),Integer(1),Integer(1))))
[]

Using a Polyomino as input:

Sage

sage: b = Polyomino([(0,0), (0,1), (0,2), (1,0), (1,1), (1,2)])
sage: p = Polyomino([(0,0)])
sage: list(p.translated_copies(b))
[Polyomino: [(0, 0)], Color: gray,
 Polyomino: [(0, 1)], Color: gray,
 Polyomino: [(0, 2)], Color: gray,
 Polyomino: [(1, 0)], Color: gray,
 Polyomino: [(1, 1)], Color: gray,
 Polyomino: [(1, 2)], Color: gray]

Python

>>> from sage.all import *
>>> b = Polyomino([(Integer(0),Integer(0)), (Integer(0),Integer(1)), (Integer(0),Integer(2)), (Integer(1),Integer(0)), (Integer(1),Integer(1)), (Integer(1),Integer(2))])
>>> p = Polyomino([(Integer(0),Integer(0))])
>>> list(p.translated_copies(b))
[Polyomino: [(0, 0)], Color: gray,
 Polyomino: [(0, 1)], Color: gray,
 Polyomino: [(0, 2)], Color: gray,
 Polyomino: [(1, 0)], Color: gray,
 Polyomino: [(1, 1)], Color: gray,
 Polyomino: [(1, 2)], Color: gray]

Sage

sage: p = Polyomino([(0,0), (1,0), (0,1)])
sage: b = Polyomino([(0,0), (1,0), (2,0), (0,1), (1,1), (0,2)])
sage: list(p.translated_copies(b))
[Polyomino: [(0, 0), (0, 1), (1, 0)], Color: gray,
 Polyomino: [(0, 1), (0, 2), (1, 1)], Color: gray,
 Polyomino: [(1, 0), (1, 1), (2, 0)], Color: gray]

Python

>>> from sage.all import *
>>> p = Polyomino([(Integer(0),Integer(0)), (Integer(1),Integer(0)), (Integer(0),Integer(1))])
>>> b = Polyomino([(Integer(0),Integer(0)), (Integer(1),Integer(0)), (Integer(2),Integer(0)), (Integer(0),Integer(1)), (Integer(1),Integer(1)), (Integer(0),Integer(2))])
>>> list(p.translated_copies(b))
[Polyomino: [(0, 0), (0, 1), (1, 0)], Color: gray,
 Polyomino: [(0, 1), (0, 2), (1, 1)], Color: gray,
 Polyomino: [(1, 0), (1, 1), (2, 0)], Color: gray]

translated_copies_intersection(box)

Return the set of non empty intersections of translated images of self with a polyomino.

INPUT:

• box – polyomino or tuple of integers (size of a box)

OUTPUT: set of 3d polyominoes

EXAMPLES:

Sage

sage: from sage.combinat.tiling import Polyomino
sage: p = Polyomino([(0,0),(1,0)], color='deeppink')
sage: sorted(sorted(a.frozenset())
....:        for a in p.translated_copies_intersection(box=(2,3)))
[[(0, 0)],
 [(0, 0), (1, 0)],
 [(0, 1)],
 [(0, 1), (1, 1)],
 [(0, 2)],
 [(0, 2), (1, 2)],
 [(1, 0)],
 [(1, 1)],
 [(1, 2)]]

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import Polyomino
>>> p = Polyomino([(Integer(0),Integer(0)),(Integer(1),Integer(0))], color='deeppink')
>>> sorted(sorted(a.frozenset())
...        for a in p.translated_copies_intersection(box=(Integer(2),Integer(3))))
[[(0, 0)],
 [(0, 0), (1, 0)],
 [(0, 1)],
 [(0, 1), (1, 1)],
 [(0, 2)],
 [(0, 2), (1, 2)],
 [(1, 0)],
 [(1, 1)],
 [(1, 2)]]

Using a Polyomino as input:

Sage

sage: b = Polyomino([(0,0), (0,1), (0,2), (1,0), (2,0)])
sage: p = Polyomino([(0,0), (1,0)])
sage: sorted(sorted(a.frozenset())
....:        for a in p.translated_copies_intersection(b))
[[(0, 0)], [(0, 0), (1, 0)], [(0, 1)], [(0, 2)], [(1, 0), (2, 0)], [(2, 0)]]

Python

>>> from sage.all import *
>>> b = Polyomino([(Integer(0),Integer(0)), (Integer(0),Integer(1)), (Integer(0),Integer(2)), (Integer(1),Integer(0)), (Integer(2),Integer(0))])
>>> p = Polyomino([(Integer(0),Integer(0)), (Integer(1),Integer(0))])
>>> sorted(sorted(a.frozenset())
...        for a in p.translated_copies_intersection(b))
[[(0, 0)], [(0, 0), (1, 0)], [(0, 1)], [(0, 2)], [(1, 0), (2, 0)], [(2, 0)]]

class sage.combinat.tiling.TilingSolver(pieces, box, rotation=True, reflection=False, reusable=False, outside=False)

Bases: SageObject

Tiling solver.

Solve the problem of tiling a polyomino with a certain number of polyominoes.

INPUT:

• pieces – iterable of Polyominoes

• box – polyomino or tuple of integers (size of a box)

• rotation – boolean (default: True); whether to allow rotations

• reflection – boolean (default: False); whether to allow reflections

• reusable – boolean (default: False); whether to allow the pieces to be reused

• outside – boolean (default: False); whether to allow pieces to partially go outside of the box (all non-empty intersection of the pieces with the box are considered)

EXAMPLES:

By default, rotations are allowed and reflections are not allowed:

Sage

sage: from sage.combinat.tiling import TilingSolver, Polyomino
sage: p = Polyomino([(0,0,0)])
sage: q = Polyomino([(0,0,0), (0,0,1)])
sage: r = Polyomino([(0,0,0), (0,0,1), (0,0,2)])
sage: T = TilingSolver([p,q,r], box=(1,1,6))
sage: T
Tiling solver of 3 pieces into a box of size 6
Rotation allowed: True
Reflection allowed: False
Reusing pieces allowed: False

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import TilingSolver, Polyomino
>>> p = Polyomino([(Integer(0),Integer(0),Integer(0))])
>>> q = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(0),Integer(1))])
>>> r = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(0),Integer(1)), (Integer(0),Integer(0),Integer(2))])
>>> T = TilingSolver([p,q,r], box=(Integer(1),Integer(1),Integer(6)))
>>> T
Tiling solver of 3 pieces into a box of size 6
Rotation allowed: True
Reflection allowed: False
Reusing pieces allowed: False

Solutions are given by an iterator:

Sage

sage: it = T.solve()
sage: for p in next(it): p
Polyomino: [(0, 0, 0)], Color: gray
Polyomino: [(0, 0, 1), (0, 0, 2)], Color: gray
Polyomino: [(0, 0, 3), (0, 0, 4), (0, 0, 5)], Color: gray

Python

>>> from sage.all import *
>>> it = T.solve()
>>> for p in next(it): p
Polyomino: [(0, 0, 0)], Color: gray
Polyomino: [(0, 0, 1), (0, 0, 2)], Color: gray
Polyomino: [(0, 0, 3), (0, 0, 4), (0, 0, 5)], Color: gray

Another solution:

Sage

sage: for p in next(it): p
Polyomino: [(0, 0, 0)], Color: gray
Polyomino: [(0, 0, 1), (0, 0, 2), (0, 0, 3)], Color: gray
Polyomino: [(0, 0, 4), (0, 0, 5)], Color: gray

Python

>>> from sage.all import *
>>> for p in next(it): p
Polyomino: [(0, 0, 0)], Color: gray
Polyomino: [(0, 0, 1), (0, 0, 2), (0, 0, 3)], Color: gray
Polyomino: [(0, 0, 4), (0, 0, 5)], Color: gray

Tiling of a polyomino by polyominoes:

Sage

sage: b = Polyomino([(0,0), (1,0), (1,1), (2,1), (1,2), (2,2), (0,3), (1,3)])
sage: p = Polyomino([(0,0), (1,0)])
sage: T = TilingSolver([p], box=b, reusable=True)
sage: T.number_of_solutions()
2

Python

>>> from sage.all import *
>>> b = Polyomino([(Integer(0),Integer(0)), (Integer(1),Integer(0)), (Integer(1),Integer(1)), (Integer(2),Integer(1)), (Integer(1),Integer(2)), (Integer(2),Integer(2)), (Integer(0),Integer(3)), (Integer(1),Integer(3))])
>>> p = Polyomino([(Integer(0),Integer(0)), (Integer(1),Integer(0))])
>>> T = TilingSolver([p], box=b, reusable=True)
>>> T.number_of_solutions()
2

animate(partial=None, stop=None, size=0.75, axes=False)

Return an animation of evolving solutions.

INPUT:

• partial – string (default: None); whether to include partial (incomplete) solutions. It can be one of the following:

  • None – include only complete solutions

  • 'common_prefix' – common prefix between two consecutive solutions

  • 'incremental' – one piece change at a time

• stop – integer (default: None); number of frames

• size – number (default: 0.75); the size of each 1 \times 1 square. This does a homothety with respect to the center of each polyomino.

• axes – boolean (default: False); whether the x and y axes are shown

EXAMPLES:

Sage

sage: from sage.combinat.tiling import Polyomino, TilingSolver
sage: y = Polyomino([(0,0),(1,0),(2,0),(3,0),(2,1)], color='cyan')
sage: T = TilingSolver([y], box=(5,10), reusable=True, reflection=True)
sage: a = T.animate()                                                       # needs sage.plot
sage: a                             # long time, optional - imagemagick, needs sage.plot
Animation with 10 frames

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import Polyomino, TilingSolver
>>> y = Polyomino([(Integer(0),Integer(0)),(Integer(1),Integer(0)),(Integer(2),Integer(0)),(Integer(3),Integer(0)),(Integer(2),Integer(1))], color='cyan')
>>> T = TilingSolver([y], box=(Integer(5),Integer(10)), reusable=True, reflection=True)
>>> a = T.animate()                                                       # needs sage.plot
>>> a                             # long time, optional - imagemagick, needs sage.plot
Animation with 10 frames

Include partial solutions (common prefix between two consecutive solutions):

Sage

sage: a = T.animate('common_prefix')                                        # needs sage.plot
sage: a                             # long time, optional - imagemagick, needs sage.plot
Animation with 19 frames

Python

>>> from sage.all import *
>>> a = T.animate('common_prefix')                                        # needs sage.plot
>>> a                             # long time, optional - imagemagick, needs sage.plot
Animation with 19 frames

Incremental solutions (one piece removed or added at a time):

Sage

sage: a = T.animate('incremental')  # long time (2s)                        # needs sage.plot
sage: a                             # long time (2s), optional - imagemagick, needs sage.plot
Animation with 123 frames

Python

>>> from sage.all import *
>>> a = T.animate('incremental')  # long time (2s)                        # needs sage.plot
>>> a                             # long time (2s), optional - imagemagick, needs sage.plot
Animation with 123 frames

Sage

sage: a.show()                      # long time, optional - imagemagick, needs sage.plot

Python

>>> from sage.all import *
>>> a.show()                      # long time, optional - imagemagick, needs sage.plot

The show function takes arguments to specify the delay between frames (measured in hundredths of a second, default value 20) and the number of iterations (default: 0, which means to iterate forever). To iterate 4 times with half a second between each frame:

Sage

sage: a.show(delay=50, iterations=4)        # long time, optional - imagemagick, needs sage.plot

Python

>>> from sage.all import *
>>> a.show(delay=Integer(50), iterations=Integer(4))        # long time, optional - imagemagick, needs sage.plot

Limit the number of frames:

Sage

sage: a = T.animate('incremental', stop=13); a      # not tested            # needs sage.plot
Animation with 13 frames

Python

>>> from sage.all import *
>>> a = T.animate('incremental', stop=Integer(13)); a      # not tested            # needs sage.plot
Animation with 13 frames

coord_to_int_dict()

Return a dictionary mapping coordinates to integers.

OUTPUT: dictionary

EXAMPLES:

Sage

sage: from sage.combinat.tiling import TilingSolver, Polyomino
sage: p = Polyomino([(0,0,0)])
sage: q = Polyomino([(0,0,0), (0,0,1)])
sage: r = Polyomino([(0,0,0), (0,0,1), (0,0,2)])
sage: T = TilingSolver([p,q,r], box=(1,1,6))
sage: A = T.coord_to_int_dict()
sage: sorted(A.items())
[((0, 0, 0), 3), ((0, 0, 1), 4), ((0, 0, 2), 5),
 ((0, 0, 3), 6), ((0, 0, 4), 7), ((0, 0, 5), 8)]

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import TilingSolver, Polyomino
>>> p = Polyomino([(Integer(0),Integer(0),Integer(0))])
>>> q = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(0),Integer(1))])
>>> r = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(0),Integer(1)), (Integer(0),Integer(0),Integer(2))])
>>> T = TilingSolver([p,q,r], box=(Integer(1),Integer(1),Integer(6)))
>>> A = T.coord_to_int_dict()
>>> sorted(A.items())
[((0, 0, 0), 3), ((0, 0, 1), 4), ((0, 0, 2), 5),
 ((0, 0, 3), 6), ((0, 0, 4), 7), ((0, 0, 5), 8)]

Reusable pieces:

Sage

sage: p = Polyomino([(0,0), (0,1)])
sage: q = Polyomino([(0,0), (0,1), (1,0), (1,1)])
sage: T = TilingSolver([p,q], box=[3,2], reusable=True)
sage: B = T.coord_to_int_dict()
sage: sorted(B.items())
[((0, 0), 0), ((0, 1), 1), ((1, 0), 2), ((1, 1), 3),
 ((2, 0), 4), ((2, 1), 5)]

Python

>>> from sage.all import *
>>> p = Polyomino([(Integer(0),Integer(0)), (Integer(0),Integer(1))])
>>> q = Polyomino([(Integer(0),Integer(0)), (Integer(0),Integer(1)), (Integer(1),Integer(0)), (Integer(1),Integer(1))])
>>> T = TilingSolver([p,q], box=[Integer(3),Integer(2)], reusable=True)
>>> B = T.coord_to_int_dict()
>>> sorted(B.items())
[((0, 0), 0), ((0, 1), 1), ((1, 0), 2), ((1, 1), 3),
 ((2, 0), 4), ((2, 1), 5)]

dlx_solver()

Return the sage DLX solver of that tiling problem.

OUTPUT: dLX Solver

EXAMPLES:

Sage

sage: from sage.combinat.tiling import TilingSolver, Polyomino
sage: p = Polyomino([(0,0,0)])
sage: q = Polyomino([(0,0,0), (0,0,1)])
sage: r = Polyomino([(0,0,0), (0,0,1), (0,0,2)])
sage: T = TilingSolver([p,q,r], box=(1,1,6))
sage: T.dlx_solver()
Dancing links solver for 9 columns and 15 rows

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import TilingSolver, Polyomino
>>> p = Polyomino([(Integer(0),Integer(0),Integer(0))])
>>> q = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(0),Integer(1))])
>>> r = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(0),Integer(1)), (Integer(0),Integer(0),Integer(2))])
>>> T = TilingSolver([p,q,r], box=(Integer(1),Integer(1),Integer(6)))
>>> T.dlx_solver()
Dancing links solver for 9 columns and 15 rows

int_to_coord_dict()

Return a dictionary mapping integers to coordinates.

EXAMPLES:

Sage

sage: from sage.combinat.tiling import TilingSolver, Polyomino
sage: p = Polyomino([(0,0,0)])
sage: q = Polyomino([(0,0,0), (0,0,1)])
sage: r = Polyomino([(0,0,0), (0,0,1), (0,0,2)])
sage: T = TilingSolver([p,q,r], box=(1,1,6))
sage: B = T.int_to_coord_dict()
sage: sorted(B.items())
[(3, (0, 0, 0)), (4, (0, 0, 1)), (5, (0, 0, 2)),
 (6, (0, 0, 3)), (7, (0, 0, 4)), (8, (0, 0, 5))]

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import TilingSolver, Polyomino
>>> p = Polyomino([(Integer(0),Integer(0),Integer(0))])
>>> q = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(0),Integer(1))])
>>> r = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(0),Integer(1)), (Integer(0),Integer(0),Integer(2))])
>>> T = TilingSolver([p,q,r], box=(Integer(1),Integer(1),Integer(6)))
>>> B = T.int_to_coord_dict()
>>> sorted(B.items())
[(3, (0, 0, 0)), (4, (0, 0, 1)), (5, (0, 0, 2)),
 (6, (0, 0, 3)), (7, (0, 0, 4)), (8, (0, 0, 5))]

Reusable pieces:

Sage

sage: from sage.combinat.tiling import Polyomino, TilingSolver
sage: p = Polyomino([(0,0), (0,1)])
sage: q = Polyomino([(0,0), (0,1), (1,0), (1,1)])
sage: T = TilingSolver([p,q], box=[3,2], reusable=True)
sage: B = T.int_to_coord_dict()
sage: sorted(B.items())
[(0, (0, 0)), (1, (0, 1)), (2, (1, 0)),
 (3, (1, 1)), (4, (2, 0)), (5, (2, 1))]

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import Polyomino, TilingSolver
>>> p = Polyomino([(Integer(0),Integer(0)), (Integer(0),Integer(1))])
>>> q = Polyomino([(Integer(0),Integer(0)), (Integer(0),Integer(1)), (Integer(1),Integer(0)), (Integer(1),Integer(1))])
>>> T = TilingSolver([p,q], box=[Integer(3),Integer(2)], reusable=True)
>>> B = T.int_to_coord_dict()
>>> sorted(B.items())
[(0, (0, 0)), (1, (0, 1)), (2, (1, 0)),
 (3, (1, 1)), (4, (2, 0)), (5, (2, 1))]

is_suitable()

Return whether the volume of the box is equal to sum of the volume of the polyominoes and the number of rows sent to the DLX solver is larger than zero.

If these conditions are not verified, then the problem is not suitable in the sense that there are no solution.

EXAMPLES:

Sage

sage: from sage.combinat.tiling import TilingSolver, Polyomino
sage: p = Polyomino([(0,0,0)])
sage: q = Polyomino([(0,0,0), (0,0,1)])
sage: r = Polyomino([(0,0,0), (0,0,1), (0,0,2)])
sage: T = TilingSolver([p,q,r], box=(1,1,6))
sage: T.is_suitable()
True
sage: T = TilingSolver([p,q,r], box=(1,1,7))
sage: T.is_suitable()
False

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import TilingSolver, Polyomino
>>> p = Polyomino([(Integer(0),Integer(0),Integer(0))])
>>> q = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(0),Integer(1))])
>>> r = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(0),Integer(1)), (Integer(0),Integer(0),Integer(2))])
>>> T = TilingSolver([p,q,r], box=(Integer(1),Integer(1),Integer(6)))
>>> T.is_suitable()
True
>>> T = TilingSolver([p,q,r], box=(Integer(1),Integer(1),Integer(7)))
>>> T.is_suitable()
False

nrows_per_piece()

Return the number of rows necessary by each piece.

OUTPUT: list

EXAMPLES:

Sage

sage: from sage.games.quantumino import QuantuminoSolver
sage: q = QuantuminoSolver(0)
sage: T = q.tiling_solver()
sage: T.nrows_per_piece()             # long time (10s)
[360, 360, 360, 360, 360, 180, 180, 672, 672, 360, 360, 180, 180, 360, 360, 180]

Python

>>> from sage.all import *
>>> from sage.games.quantumino import QuantuminoSolver
>>> q = QuantuminoSolver(Integer(0))
>>> T = q.tiling_solver()
>>> T.nrows_per_piece()             # long time (10s)
[360, 360, 360, 360, 360, 180, 180, 672, 672, 360, 360, 180, 180, 360, 360, 180]

number_of_solutions()

Return the number of distinct solutions.

OUTPUT: integer

EXAMPLES:

Sage

sage: from sage.combinat.tiling import TilingSolver, Polyomino
sage: p = Polyomino([(0,0)])
sage: q = Polyomino([(0,0), (0,1)])
sage: r = Polyomino([(0,0), (0,1), (0,2)])
sage: T = TilingSolver([p,q,r], box=(1,6))
sage: T.number_of_solutions()
6

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import TilingSolver, Polyomino
>>> p = Polyomino([(Integer(0),Integer(0))])
>>> q = Polyomino([(Integer(0),Integer(0)), (Integer(0),Integer(1))])
>>> r = Polyomino([(Integer(0),Integer(0)), (Integer(0),Integer(1)), (Integer(0),Integer(2))])
>>> T = TilingSolver([p,q,r], box=(Integer(1),Integer(6)))
>>> T.number_of_solutions()
6

Sage

sage: T = TilingSolver([p,q,r], box=(1,7))
sage: T.number_of_solutions()
0

Python

>>> from sage.all import *
>>> T = TilingSolver([p,q,r], box=(Integer(1),Integer(7)))
>>> T.number_of_solutions()
0

pieces()

Return the list of pieces.

OUTPUT: list of 3d polyominoes

EXAMPLES:

Sage

sage: from sage.combinat.tiling import TilingSolver, Polyomino
sage: p = Polyomino([(0,0,0)])
sage: q = Polyomino([(0,0,0), (0,0,1)])
sage: r = Polyomino([(0,0,0), (0,0,1), (0,0,2)])
sage: T = TilingSolver([p,q,r], box=(1,1,6))
sage: for p in T._pieces: p
Polyomino: [(0, 0, 0)], Color: gray
Polyomino: [(0, 0, 0), (0, 0, 1)], Color: gray
Polyomino: [(0, 0, 0), (0, 0, 1), (0, 0, 2)], Color: gray

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import TilingSolver, Polyomino
>>> p = Polyomino([(Integer(0),Integer(0),Integer(0))])
>>> q = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(0),Integer(1))])
>>> r = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(0),Integer(1)), (Integer(0),Integer(0),Integer(2))])
>>> T = TilingSolver([p,q,r], box=(Integer(1),Integer(1),Integer(6)))
>>> for p in T._pieces: p
Polyomino: [(0, 0, 0)], Color: gray
Polyomino: [(0, 0, 0), (0, 0, 1)], Color: gray
Polyomino: [(0, 0, 0), (0, 0, 1), (0, 0, 2)], Color: gray

row_to_polyomino(row_number)

Return a polyomino associated to a row.

INPUT:

• row_number – integer; the \(i\)-th row

OUTPUT: polyomino

EXAMPLES:

Sage

sage: from sage.combinat.tiling import TilingSolver, Polyomino
sage: a = Polyomino([(0,0,0), (0,0,1), (1,0,0)], color='blue')
sage: b = Polyomino([(0,0,0), (1,0,0), (0,1,0)], color='red')
sage: T = TilingSolver([a,b], box=(2,1,3))
sage: len(T.rows())
16

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import TilingSolver, Polyomino
>>> a = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(0),Integer(1)), (Integer(1),Integer(0),Integer(0))], color='blue')
>>> b = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(1),Integer(0),Integer(0)), (Integer(0),Integer(1),Integer(0))], color='red')
>>> T = TilingSolver([a,b], box=(Integer(2),Integer(1),Integer(3)))
>>> len(T.rows())
16

Sage

sage: T.row_to_polyomino(7)
Polyomino: [(0, 0, 2), (1, 0, 1), (1, 0, 2)], Color: blue

Python

>>> from sage.all import *
>>> T.row_to_polyomino(Integer(7))
Polyomino: [(0, 0, 2), (1, 0, 1), (1, 0, 2)], Color: blue

Sage

sage: T.row_to_polyomino(13)
Polyomino: [(0, 0, 1), (1, 0, 1), (1, 0, 2)], Color: red

Python

>>> from sage.all import *
>>> T.row_to_polyomino(Integer(13))
Polyomino: [(0, 0, 1), (1, 0, 1), (1, 0, 2)], Color: red

rows()

Creation of the rows.

EXAMPLES:

Sage

sage: from sage.combinat.tiling import TilingSolver, Polyomino
sage: p = Polyomino([(0,0,0)])
sage: q = Polyomino([(0,0,0), (0,0,1)])
sage: r = Polyomino([(0,0,0), (0,0,1), (0,0,2)])
sage: T = TilingSolver([p,q,r], box=(1,1,6))
sage: rows = T.rows()
sage: for row in rows: row
[0, 3]
[0, 4]
[0, 5]
[0, 6]
[0, 7]
[0, 8]
[1, 3, 4]
[1, 4, 5]
[1, 5, 6]
[1, 6, 7]
[1, 7, 8]
[2, 3, 4, 5]
[2, 4, 5, 6]
[2, 5, 6, 7]
[2, 6, 7, 8]

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import TilingSolver, Polyomino
>>> p = Polyomino([(Integer(0),Integer(0),Integer(0))])
>>> q = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(0),Integer(1))])
>>> r = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(0),Integer(1)), (Integer(0),Integer(0),Integer(2))])
>>> T = TilingSolver([p,q,r], box=(Integer(1),Integer(1),Integer(6)))
>>> rows = T.rows()
>>> for row in rows: row
[0, 3]
[0, 4]
[0, 5]
[0, 6]
[0, 7]
[0, 8]
[1, 3, 4]
[1, 4, 5]
[1, 5, 6]
[1, 6, 7]
[1, 7, 8]
[2, 3, 4, 5]
[2, 4, 5, 6]
[2, 5, 6, 7]
[2, 6, 7, 8]

rows_for_piece(i, mod_box_isometries=False)

Return the rows for the \(i\)-th piece.

INPUT:

• i – integer; the \(i\)-th piece

• mod_box_isometries – boolean (default: False); whether to consider only rows for positions up to the action of the quotient the group of isometries of the \(n\)-cube by the subgroup of isometries of the \(a_1\times a_2\cdots \times a_n\) rectangular box where are the \(a_i\) are assumed to be distinct.

EXAMPLES:

Sage

sage: from sage.combinat.tiling import TilingSolver, Polyomino
sage: p = Polyomino([(0,0,0)])
sage: q = Polyomino([(0,0,0), (0,0,1)])
sage: r = Polyomino([(0,0,0), (0,0,1), (0,0,2)])
sage: T = TilingSolver([p,q,r], box=(1,1,6))
sage: T.rows_for_piece(0)
[[0, 3], [0, 4], [0, 5], [0, 6], [0, 7], [0, 8]]
sage: T.rows_for_piece(1)
[[1, 3, 4], [1, 4, 5], [1, 5, 6], [1, 6, 7], [1, 7, 8]]
sage: T.rows_for_piece(2)
[[2, 3, 4, 5], [2, 4, 5, 6], [2, 5, 6, 7], [2, 6, 7, 8]]

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import TilingSolver, Polyomino
>>> p = Polyomino([(Integer(0),Integer(0),Integer(0))])
>>> q = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(0),Integer(1))])
>>> r = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(0),Integer(1)), (Integer(0),Integer(0),Integer(2))])
>>> T = TilingSolver([p,q,r], box=(Integer(1),Integer(1),Integer(6)))
>>> T.rows_for_piece(Integer(0))
[[0, 3], [0, 4], [0, 5], [0, 6], [0, 7], [0, 8]]
>>> T.rows_for_piece(Integer(1))
[[1, 3, 4], [1, 4, 5], [1, 5, 6], [1, 6, 7], [1, 7, 8]]
>>> T.rows_for_piece(Integer(2))
[[2, 3, 4, 5], [2, 4, 5, 6], [2, 5, 6, 7], [2, 6, 7, 8]]

Less rows when using mod_box_isometries=True:

Sage

sage: a = Polyomino([(0,0,0), (0,0,1), (1,0,0)])
sage: b = Polyomino([(0,0,0), (1,0,0), (0,1,0)])
sage: T = TilingSolver([a,b], box=(2,1,3))
sage: T.rows_for_piece(0)
[[0, 2, 3, 5],
 [0, 3, 4, 6],
 [0, 2, 3, 6],
 [0, 3, 4, 7],
 [0, 2, 5, 6],
 [0, 3, 6, 7],
 [0, 3, 5, 6],
 [0, 4, 6, 7]]
sage: T.rows_for_piece(0, mod_box_isometries=True)
[[0, 2, 3, 5], [0, 3, 4, 6]]
sage: T.rows_for_piece(1, mod_box_isometries=True)
[[1, 2, 3, 5], [1, 3, 4, 6]]

Python

>>> from sage.all import *
>>> a = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(0),Integer(1)), (Integer(1),Integer(0),Integer(0))])
>>> b = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(1),Integer(0),Integer(0)), (Integer(0),Integer(1),Integer(0))])
>>> T = TilingSolver([a,b], box=(Integer(2),Integer(1),Integer(3)))
>>> T.rows_for_piece(Integer(0))
[[0, 2, 3, 5],
 [0, 3, 4, 6],
 [0, 2, 3, 6],
 [0, 3, 4, 7],
 [0, 2, 5, 6],
 [0, 3, 6, 7],
 [0, 3, 5, 6],
 [0, 4, 6, 7]]
>>> T.rows_for_piece(Integer(0), mod_box_isometries=True)
[[0, 2, 3, 5], [0, 3, 4, 6]]
>>> T.rows_for_piece(Integer(1), mod_box_isometries=True)
[[1, 2, 3, 5], [1, 3, 4, 6]]

solve(partial=None)

Return an iterator of list of polyominoes that are an exact cover of the box.

INPUT:

• partial – string (default: None); whether to include partial (incomplete) solutions. It can be one of the following:

  • None – include only complete solution

  • 'common_prefix' – common prefix between two consecutive solutions

  • 'incremental' – one piece change at a time

OUTPUT: iterator of list of polyominoes

EXAMPLES:

Sage

sage: from sage.combinat.tiling import TilingSolver, Polyomino
sage: p = Polyomino([(0,0,0)])
sage: q = Polyomino([(0,0,0), (0,0,1)])
sage: r = Polyomino([(0,0,0), (0,0,1), (0,0,2)])
sage: T = TilingSolver([p,q,r], box=(1,1,6))
sage: it = T.solve()
sage: for p in next(it): p
Polyomino: [(0, 0, 0)], Color: gray
Polyomino: [(0, 0, 1), (0, 0, 2)], Color: gray
Polyomino: [(0, 0, 3), (0, 0, 4), (0, 0, 5)], Color: gray
sage: for p in next(it): p
Polyomino: [(0, 0, 0)], Color: gray
Polyomino: [(0, 0, 1), (0, 0, 2), (0, 0, 3)], Color: gray
Polyomino: [(0, 0, 4), (0, 0, 5)], Color: gray
sage: for p in next(it): p
Polyomino: [(0, 0, 0), (0, 0, 1)], Color: gray
Polyomino: [(0, 0, 2), (0, 0, 3), (0, 0, 4)], Color: gray
Polyomino: [(0, 0, 5)], Color: gray

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import TilingSolver, Polyomino
>>> p = Polyomino([(Integer(0),Integer(0),Integer(0))])
>>> q = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(0),Integer(1))])
>>> r = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(0),Integer(1)), (Integer(0),Integer(0),Integer(2))])
>>> T = TilingSolver([p,q,r], box=(Integer(1),Integer(1),Integer(6)))
>>> it = T.solve()
>>> for p in next(it): p
Polyomino: [(0, 0, 0)], Color: gray
Polyomino: [(0, 0, 1), (0, 0, 2)], Color: gray
Polyomino: [(0, 0, 3), (0, 0, 4), (0, 0, 5)], Color: gray
>>> for p in next(it): p
Polyomino: [(0, 0, 0)], Color: gray
Polyomino: [(0, 0, 1), (0, 0, 2), (0, 0, 3)], Color: gray
Polyomino: [(0, 0, 4), (0, 0, 5)], Color: gray
>>> for p in next(it): p
Polyomino: [(0, 0, 0), (0, 0, 1)], Color: gray
Polyomino: [(0, 0, 2), (0, 0, 3), (0, 0, 4)], Color: gray
Polyomino: [(0, 0, 5)], Color: gray

Including the partial solutions:

Sage

sage: it = T.solve(partial='common_prefix')
sage: for p in next(it): p
Polyomino: [(0, 0, 0)], Color: gray
Polyomino: [(0, 0, 1), (0, 0, 2)], Color: gray
Polyomino: [(0, 0, 3), (0, 0, 4), (0, 0, 5)], Color: gray
sage: for p in next(it): p
Polyomino: [(0, 0, 0)], Color: gray
sage: for p in next(it): p
Polyomino: [(0, 0, 0)], Color: gray
Polyomino: [(0, 0, 1), (0, 0, 2), (0, 0, 3)], Color: gray
Polyomino: [(0, 0, 4), (0, 0, 5)], Color: gray
sage: for p in next(it): p
sage: for p in next(it): p
Polyomino: [(0, 0, 0), (0, 0, 1)], Color: gray
Polyomino: [(0, 0, 2), (0, 0, 3), (0, 0, 4)], Color: gray
Polyomino: [(0, 0, 5)], Color: gray

Python

>>> from sage.all import *
>>> it = T.solve(partial='common_prefix')
>>> for p in next(it): p
Polyomino: [(0, 0, 0)], Color: gray
Polyomino: [(0, 0, 1), (0, 0, 2)], Color: gray
Polyomino: [(0, 0, 3), (0, 0, 4), (0, 0, 5)], Color: gray
>>> for p in next(it): p
Polyomino: [(0, 0, 0)], Color: gray
>>> for p in next(it): p
Polyomino: [(0, 0, 0)], Color: gray
Polyomino: [(0, 0, 1), (0, 0, 2), (0, 0, 3)], Color: gray
Polyomino: [(0, 0, 4), (0, 0, 5)], Color: gray
>>> for p in next(it): p
>>> for p in next(it): p
Polyomino: [(0, 0, 0), (0, 0, 1)], Color: gray
Polyomino: [(0, 0, 2), (0, 0, 3), (0, 0, 4)], Color: gray
Polyomino: [(0, 0, 5)], Color: gray

Colors are preserved when the polyomino can be reused:

Sage

sage: p = Polyomino([(0,0,0)], color='yellow')
sage: q = Polyomino([(0,0,0), (0,0,1)], color='yellow')
sage: r = Polyomino([(0,0,0), (0,0,1), (0,0,2)], color='yellow')
sage: T = TilingSolver([p,q,r], box=(1,1,6), reusable=True)
sage: it = T.solve()
sage: for p in next(it): p
Polyomino: [(0, 0, 0)], Color: yellow
Polyomino: [(0, 0, 1)], Color: yellow
Polyomino: [(0, 0, 2)], Color: yellow
Polyomino: [(0, 0, 3)], Color: yellow
Polyomino: [(0, 0, 4)], Color: yellow
Polyomino: [(0, 0, 5)], Color: yellow

Python

>>> from sage.all import *
>>> p = Polyomino([(Integer(0),Integer(0),Integer(0))], color='yellow')
>>> q = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(0),Integer(1))], color='yellow')
>>> r = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(0),Integer(1)), (Integer(0),Integer(0),Integer(2))], color='yellow')
>>> T = TilingSolver([p,q,r], box=(Integer(1),Integer(1),Integer(6)), reusable=True)
>>> it = T.solve()
>>> for p in next(it): p
Polyomino: [(0, 0, 0)], Color: yellow
Polyomino: [(0, 0, 1)], Color: yellow
Polyomino: [(0, 0, 2)], Color: yellow
Polyomino: [(0, 0, 3)], Color: yellow
Polyomino: [(0, 0, 4)], Color: yellow
Polyomino: [(0, 0, 5)], Color: yellow

space()

Return an iterator over all the nonnegative integer coordinates contained in the space to tile.

EXAMPLES:

Sage

sage: from sage.combinat.tiling import TilingSolver, Polyomino
sage: p = Polyomino([(0,0,0)])
sage: q = Polyomino([(0,0,0), (0,0,1)])
sage: r = Polyomino([(0,0,0), (0,0,1), (0,0,2)])
sage: T = TilingSolver([p,q,r], box=(1,1,6))
sage: list(T.space())
[(0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 0, 3), (0, 0, 4), (0, 0, 5)]

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import TilingSolver, Polyomino
>>> p = Polyomino([(Integer(0),Integer(0),Integer(0))])
>>> q = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(0),Integer(1))])
>>> r = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(0),Integer(1)), (Integer(0),Integer(0),Integer(2))])
>>> T = TilingSolver([p,q,r], box=(Integer(1),Integer(1),Integer(6)))
>>> list(T.space())
[(0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 0, 3), (0, 0, 4), (0, 0, 5)]

starting_rows()

Return the starting rows for each piece.

EXAMPLES:

Sage

sage: from sage.combinat.tiling import TilingSolver, Polyomino
sage: p = Polyomino([(0,0,0)])
sage: q = Polyomino([(0,0,0), (0,0,1)])
sage: r = Polyomino([(0,0,0), (0,0,1), (0,0,2)])
sage: T = TilingSolver([p,q,r], box=(1,1,6))
sage: T.starting_rows()
[0, 6, 11, 15]

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import TilingSolver, Polyomino
>>> p = Polyomino([(Integer(0),Integer(0),Integer(0))])
>>> q = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(0),Integer(1))])
>>> r = Polyomino([(Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(0),Integer(1)), (Integer(0),Integer(0),Integer(2))])
>>> T = TilingSolver([p,q,r], box=(Integer(1),Integer(1),Integer(6)))
>>> T.starting_rows()
[0, 6, 11, 15]

sage.combinat.tiling.ncube_isometry_group(n, orientation_preserving=True)

Return the isometry group of the \(n\)-cube as a list of matrices.

INPUT:

• n – positive integer, dimension of the space

• orientation_preserving – boolean (default: True); whether the orientation is preserved

OUTPUT: list of matrices

EXAMPLES:

Sage

sage: from sage.combinat.tiling import ncube_isometry_group
sage: ncube_isometry_group(2)
[
[1 0]  [ 0  1]  [-1  0]  [ 0 -1]
[0 1], [-1  0], [ 0 -1], [ 1  0]
]
sage: ncube_isometry_group(2, orientation_preserving=False)
[
[1 0]  [ 0 -1]  [ 1  0]  [ 0  1]  [0 1]  [-1  0]  [ 0 -1]  [-1  0]
[0 1], [-1  0], [ 0 -1], [-1  0], [1 0], [ 0 -1], [ 1  0], [ 0  1]
]

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import ncube_isometry_group
>>> ncube_isometry_group(Integer(2))
[
[1 0]  [ 0  1]  [-1  0]  [ 0 -1]
[0 1], [-1  0], [ 0 -1], [ 1  0]
]
>>> ncube_isometry_group(Integer(2), orientation_preserving=False)
[
[1 0]  [ 0 -1]  [ 1  0]  [ 0  1]  [0 1]  [-1  0]  [ 0 -1]  [-1  0]
[0 1], [-1  0], [ 0 -1], [-1  0], [1 0], [ 0 -1], [ 1  0], [ 0  1]
]

There are 24 orientation preserving isometries of the 3-cube:

Sage

sage: ncube_isometry_group(3)
[
[1 0 0]  [ 1  0  0]  [ 1  0  0]  [ 0  1  0]  [0 1 0]  [ 0  0  1]
[0 1 0]  [ 0  0  1]  [ 0  0 -1]  [-1  0  0]  [0 0 1]  [ 0 -1  0]
[0 0 1], [ 0 -1  0], [ 0  1  0], [ 0  0  1], [1 0 0], [ 1  0  0],

[-1  0  0]  [ 0 -1  0]  [-1  0  0]  [-1  0  0]  [ 0 -1  0]  [ 0  0 -1]
[ 0 -1  0]  [ 0  0 -1]  [ 0  0 -1]  [ 0  1  0]  [ 0  0  1]  [ 1  0  0]
[ 0  0  1], [ 1  0  0], [ 0 -1  0], [ 0  0 -1], [-1  0  0], [ 0 -1  0],

[ 0  1  0]  [ 0  0  1]  [0 0 1]  [ 0 -1  0]  [ 0  0 -1]  [-1  0  0]
[ 1  0  0]  [ 0  1  0]  [1 0 0]  [ 1  0  0]  [ 0  1  0]  [ 0  0  1]
[ 0  0 -1], [-1  0  0], [0 1 0], [ 0  0  1], [ 1  0  0], [ 0  1  0],

[ 0 -1  0]  [ 0  0 -1]  [ 0  0  1]  [ 1  0  0]  [ 0  0 -1]  [ 0  1  0]
[-1  0  0]  [-1  0  0]  [-1  0  0]  [ 0 -1  0]  [ 0 -1  0]  [ 0  0 -1]
[ 0  0 -1], [ 0  1  0], [ 0 -1  0], [ 0  0 -1], [-1  0  0], [-1  0  0]
]

Python

>>> from sage.all import *
>>> ncube_isometry_group(Integer(3))
[
[1 0 0]  [ 1  0  0]  [ 1  0  0]  [ 0  1  0]  [0 1 0]  [ 0  0  1]
[0 1 0]  [ 0  0  1]  [ 0  0 -1]  [-1  0  0]  [0 0 1]  [ 0 -1  0]
[0 0 1], [ 0 -1  0], [ 0  1  0], [ 0  0  1], [1 0 0], [ 1  0  0],
<BLANKLINE>
[-1  0  0]  [ 0 -1  0]  [-1  0  0]  [-1  0  0]  [ 0 -1  0]  [ 0  0 -1]
[ 0 -1  0]  [ 0  0 -1]  [ 0  0 -1]  [ 0  1  0]  [ 0  0  1]  [ 1  0  0]
[ 0  0  1], [ 1  0  0], [ 0 -1  0], [ 0  0 -1], [-1  0  0], [ 0 -1  0],
<BLANKLINE>
[ 0  1  0]  [ 0  0  1]  [0 0 1]  [ 0 -1  0]  [ 0  0 -1]  [-1  0  0]
[ 1  0  0]  [ 0  1  0]  [1 0 0]  [ 1  0  0]  [ 0  1  0]  [ 0  0  1]
[ 0  0 -1], [-1  0  0], [0 1 0], [ 0  0  1], [ 1  0  0], [ 0  1  0],
<BLANKLINE>
[ 0 -1  0]  [ 0  0 -1]  [ 0  0  1]  [ 1  0  0]  [ 0  0 -1]  [ 0  1  0]
[-1  0  0]  [-1  0  0]  [-1  0  0]  [ 0 -1  0]  [ 0 -1  0]  [ 0  0 -1]
[ 0  0 -1], [ 0  1  0], [ 0 -1  0], [ 0  0 -1], [-1  0  0], [-1  0  0]
]

sage.combinat.tiling.ncube_isometry_group_cosets(orientation_preserving=True)

Return the quotient of the isometry group of the \(n\)-cube by the the isometry group of the rectangular parallelepiped.

INPUT:

• n – positive integer, dimension of the space

• orientation_preserving – boolean (default: True); whether the orientation is preserved

OUTPUT: list of cosets, each coset being a sorted list of matrices

EXAMPLES:

Sage

sage: from sage.combinat.tiling import ncube_isometry_group_cosets
sage: sorted(ncube_isometry_group_cosets(2))
[[
[-1  0]  [1 0]
[ 0 -1], [0 1]
], [
[ 0 -1]  [ 0  1]
[ 1  0], [-1  0]
]]
sage: sorted(ncube_isometry_group_cosets(2, False))
[[
[-1  0]  [-1  0]  [ 1  0]  [1 0]
[ 0 -1], [ 0  1], [ 0 -1], [0 1]
], [
[ 0 -1]  [ 0 -1]  [ 0  1]  [0 1]
[-1  0], [ 1  0], [-1  0], [1 0]
]]

Python

>>> from sage.all import *
>>> from sage.combinat.tiling import ncube_isometry_group_cosets
>>> sorted(ncube_isometry_group_cosets(Integer(2)))
[[
[-1  0]  [1 0]
[ 0 -1], [0 1]
], [
[ 0 -1]  [ 0  1]
[ 1  0], [-1  0]
]]
>>> sorted(ncube_isometry_group_cosets(Integer(2), False))
[[
[-1  0]  [-1  0]  [ 1  0]  [1 0]
[ 0 -1], [ 0  1], [ 0 -1], [0 1]
], [
[ 0 -1]  [ 0 -1]  [ 0  1]  [0 1]
[-1  0], [ 1  0], [-1  0], [1 0]
]]

Sage

sage: sorted(ncube_isometry_group_cosets(3))
[[
[-1  0  0]  [-1  0  0]  [ 1  0  0]  [1 0 0]
[ 0 -1  0]  [ 0  1  0]  [ 0 -1  0]  [0 1 0]
[ 0  0  1], [ 0  0 -1], [ 0  0 -1], [0 0 1]
], [
[-1  0  0]  [-1  0  0]  [ 1  0  0]  [ 1  0  0]
[ 0  0 -1]  [ 0  0  1]  [ 0  0 -1]  [ 0  0  1]
[ 0 -1  0], [ 0  1  0], [ 0  1  0], [ 0 -1  0]
], [
[ 0 -1  0]  [ 0 -1  0]  [ 0  1  0]  [ 0  1  0]
[-1  0  0]  [ 1  0  0]  [-1  0  0]  [ 1  0  0]
[ 0  0 -1], [ 0  0  1], [ 0  0  1], [ 0  0 -1]
], [
[ 0 -1  0]  [ 0 -1  0]  [ 0  1  0]  [0 1 0]
[ 0  0 -1]  [ 0  0  1]  [ 0  0 -1]  [0 0 1]
[ 1  0  0], [-1  0  0], [-1  0  0], [1 0 0]
], [
[ 0  0 -1]  [ 0  0 -1]  [ 0  0  1]  [0 0 1]
[-1  0  0]  [ 1  0  0]  [-1  0  0]  [1 0 0]
[ 0  1  0], [ 0 -1  0], [ 0 -1  0], [0 1 0]
], [
[ 0  0 -1]  [ 0  0 -1]  [ 0  0  1]  [ 0  0  1]
[ 0 -1  0]  [ 0  1  0]  [ 0 -1  0]  [ 0  1  0]
[-1  0  0], [ 1  0  0], [ 1  0  0], [-1  0  0]
]]

Python

>>> from sage.all import *
>>> sorted(ncube_isometry_group_cosets(Integer(3)))
[[
[-1  0  0]  [-1  0  0]  [ 1  0  0]  [1 0 0]
[ 0 -1  0]  [ 0  1  0]  [ 0 -1  0]  [0 1 0]
[ 0  0  1], [ 0  0 -1], [ 0  0 -1], [0 0 1]
], [
[-1  0  0]  [-1  0  0]  [ 1  0  0]  [ 1  0  0]
[ 0  0 -1]  [ 0  0  1]  [ 0  0 -1]  [ 0  0  1]
[ 0 -1  0], [ 0  1  0], [ 0  1  0], [ 0 -1  0]
], [
[ 0 -1  0]  [ 0 -1  0]  [ 0  1  0]  [ 0  1  0]
[-1  0  0]  [ 1  0  0]  [-1  0  0]  [ 1  0  0]
[ 0  0 -1], [ 0  0  1], [ 0  0  1], [ 0  0 -1]
], [
[ 0 -1  0]  [ 0 -1  0]  [ 0  1  0]  [0 1 0]
[ 0  0 -1]  [ 0  0  1]  [ 0  0 -1]  [0 0 1]
[ 1  0  0], [-1  0  0], [-1  0  0], [1 0 0]
], [
[ 0  0 -1]  [ 0  0 -1]  [ 0  0  1]  [0 0 1]
[-1  0  0]  [ 1  0  0]  [-1  0  0]  [1 0 0]
[ 0  1  0], [ 0 -1  0], [ 0 -1  0], [0 1 0]
], [
[ 0  0 -1]  [ 0  0 -1]  [ 0  0  1]  [ 0  0  1]
[ 0 -1  0]  [ 0  1  0]  [ 0 -1  0]  [ 0  1  0]
[-1  0  0], [ 1  0  0], [ 1  0  0], [-1  0  0]
]]